#232242

order	512
nilpotency class	3
isoclinism	isoclinic class of #232235
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1538 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1538 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,11,8)(7,14,15,20)(9,19,10,13)(12,17,18,16)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,35)(34,36);(5,7)(6,12)(8,18)(9,17)(10,16)(11,15)(13,20)(14,19)(21,23,25,30)(22,26,28,33)(24,29,31,36)(27,34,35,32);(2,4)(5,8,11,6)(7,14,15,20)(9,19,10,13)(12,16,18,17)(21,24)(22,27)(23,29)(25,31)(26,32)(28,35)(30,36)(33,34);(7,15)(12,18)(14,20)(16,17)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139438);