#93896

order	512
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#353
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	C₄⋊#95	C₂²
prime radical	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₄⋊#95 ◁ G
LECS	G ▷ C₄⋊#95 ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 429 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=f,d^2=g,e^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=defi,e^b=ei,d^c=dhi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267844
maximal normal subgroup of	#72267844
maximal quotient of	#72267847, #72267854

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1092);