#497140

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33713	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

646 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,7)(5,8)(9,10,14,19,15,21,32,20)(11,16,25,37,23,35,51,38)(12,26,29,53,30,55,57,54)(13,31,27,18,33,43,56,44)(17,39,40,58,42,65,41,28)(22,45,49,67,47,66,71,68)(24,52,46,36,50,64,69,63)(34,59,61,70,60,72,62,48);(1,3,6,2)(4,7,5,8)(9,11,14,25,15,23,32,51)(10,16,19,37,21,35,20,38)(12,22,29,49,30,47,57,71)(13,24,27,46,33,50,56,69)(17,34,40,61,42,60,41,62)(18,36,43,64,44,63,31,52)(26,45,53,67,55,66,54,68)(28,48,39,59,58,70,65,72);(1,4,6,5)(2,7,3,8)(9,12,27,39,14,29,33,58,15,30,56,65,32,57,13,28)(10,17,31,54,20,41,44,55,21,42,43,53,19,40,18,26)(11,22,46,59,25,49,50,70,23,47,69,72,51,71,24,48)(16,34,52,68,38,62,63,66,35,60,64,67,37,61,36,45);(1,5,6,4)(2,7,3,8)(11,23)(16,35)(22,47)(24,50)(25,51)(34,60)(36,63)(37,38)(45,66)(46,69)(48,70)(49,71)(52,64)(59,72)(61,62)(67,68)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404336);