#552953

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

1918 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=gh,d^2=f,e^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 460149);