#124444

order	512
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

370 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 128

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

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

Polycyclic presentation:

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

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, 31640);