#522518

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

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

1866 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)(15,16)(18,19)(21,22)(23,27)(24,31)(25,26)(28,32)(29,33)(30,36)(34,35);(1,3,4,2)(6,9)(10,12)(13,14,17,20)(15,18,16,19)(22,26)(23,29)(28,31)(30,35);(5,6,8,9)(7,10,11,12)(13,15,17,16)(14,18,20,19)(21,22)(23,27)(24,28)(25,26)(29,33)(30,34)(31,32)(35,36);(13,16,17,15)(14,19,20,18)(21,23,25,29)(22,27,26,33)(24,30,32,35)(28,34,31,36);(5,7,8,11)(6,10,9,12)(21,24)(22,28)(23,30)(25,32)(26,31)(27,34)(29,35)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgh,d^2=fg,e^2=fh,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, 429714);