#129321

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1779
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

816 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=di,f^b=fgi,e^c=ei,f^c=fi

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, 36517);