#122201

order	512
nilpotency class	3
isoclinism	isoclinic class of #97080
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1162 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1162 conjugacy classes of subgroups — too many to draw. Only its 166 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,g^b=gh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29397);