#521630

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

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

2350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fg,d^2=fgh,e^2=f,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, 428826);