#382839

order	512
nilpotency class	3
isoclinism	isoclinic class of #184805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1202 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 258 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,35)(28,37)(29,36)(30,38)(31,39)(32,40)(33,42)(34,41)(43,46)(44,47)(45,48);(2,5)(6,8)(17,19,22,29)(18,23,25,36)(20,30,31,27)(21,33,34,28)(24,38,39,35)(26,42,41,37)(32,43,45,44)(40,46,48,47);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(17,20,21,32,22,31,34,45)(18,24,26,40,25,39,41,48)(19,27,28,43,29,30,33,44)(23,35,37,46,36,38,42,47);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(18,25)(23,36)(24,39)(26,41)(35,38)(37,42)(40,48)(46,47)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=h,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,h^b=hi,d^c=de,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290035);