#102376

order	512
nilpotency class	3
isoclinism	isoclinic class of #102154
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

338 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eh,d^2=gh,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,f^b=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9572);