#525128

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=hi,d^2=fgi,e^2=gh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 432324);