#93613

order	512
nilpotency class	4
isoclinism	isoclinic class of #93565
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1286
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₄×#97	C₂²
socle	C₂	#7041
prime radical	C₂³	#344
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

159 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 159 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=fg,e^2=f,f^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,h^a=hi,c^b=cg,d^b=dh,g^b=gi,d^c=di,e^c=ei,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, 809);