#93035

order	512
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#327
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂²×C₂²⋊C₈	C₂²
socle	C₂²	#1294
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

406 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=h,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gi,e^c=ehi

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, 231);