#528910

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

1910 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)(6,7)(10,11)(13,14,17,16)(15,18,19,20)(22,26)(23,29)(24,32)(34,36);(15,19)(18,20)(21,22,25,26)(23,30,29,27)(24,28,32,33)(31,36,35,34);(1,2,4,3)(5,6,9,7)(8,10,12,11)(13,15,17,19)(14,18,16,20)(21,22)(23,27)(24,28)(25,26)(29,30)(31,34)(32,33)(35,36);(5,7,9,6)(8,11,12,10)(13,14,17,16)(15,18,19,20)(21,23,25,29)(22,27,26,30)(24,31,32,35)(28,34,33,36);(1,3,4,2)(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,20,19,18)(21,24,25,32)(22,28,26,33)(23,31,29,35)(27,34,30,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,d^2=fgh,e^2=gi,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, 436106);