#93638

order	512
nilpotency class	4
isoclinism	isoclinic class of #93520
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#342
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂×C₄×C₄⋊C₄	C₂²
prime radical	C₂⁴	#95
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₄⋊C₄ ▷ C₂⁴ ▷ C₂ ▷ C₁

342 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=fhi,e^2=h,b^a=bc,c^a=cf,e^a=efgh,c^b=cg,d^b=dh,g^b=gi,e^c=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, 834);