#124390

order	512
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	#1459
prime radical	#392	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,5)(4,6)(7,8)(9,10,12,17,13,18,25,32,14,15,24,27,26,28,35,38)(11,19,21,31,22,33,36,40,20,16,23,29,34,30,37,39);(2,5)(3,8)(6,7)(10,15)(11,20)(18,28)(21,23)(22,34)(29,31)(36,37)(39,40);(2,6)(5,7)(9,11,13,22,14,20,26,34)(10,16,18,30,15,19,28,33)(12,23,25,37,24,21,35,36)(17,31,32,40,27,29,38,39)

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,c^b=ci,d^b=dg,e^b=efg,e^c=ei,e^d=ei

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, 31586);