#92828

order	512
nilpotency class	4
isoclinism	isoclinic class of #92825
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1270
commutator	C₂×C₄²	C₂×C₈
Frattini, cosocle	#1744	C₂²
prime radical	C₂³	#323
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ #1744 ▷ C₂×C₄² ▷ C₂² ▷ C₁

184 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 42 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72210668, #72210671

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24);