#122407

order	512
nilpotency class	4
isoclinism	isoclinic class of #11753
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
socle	C₂²	#1459
prime radical	#2388	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

448 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=i,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=ci,d^b=di,e^b=efi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29603);