#93320

order	512
nilpotency class	4
isoclinism	isoclinic class of #93295
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1276
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₄×#97	C₂²
socle	C₂	#7019
prime radical	C₂³	#344
LCS	G ▷ C₄² ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ C₄×#97 ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

172 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,13,15,8,14,21,17,23)(4,10,18,25)(5,9)(6,11)(7,12)(16,27,30,32,28,31,29,22)(19,24,20,26);(1,3,4,16,6,17,20,30,7,14,18,28,5,15,19,29)(2,8,10,22,11,13,26,27,12,23,25,32,9,21,24,31)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,c^b=ci,d^b=dg,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257878, #72265522, #72265523
maximal normal subgroup of	#72257878, #72265522, #72265523

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 516);