#93560

order	512
nilpotency class	4
isoclinism	isoclinic class of #93529
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1284
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂×C₄×M₁₆	C₂²
socle	C₂	#7037
prime radical	C₂⁴	#104
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×M₁₆ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

279 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 279 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,e^2=fi,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 756);