#518673

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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

1998 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)(15,19)(18,20)(21,22)(23,27)(24,28)(25,26)(29,35)(30,33)(31,36)(32,34);(6,9)(10,12)(13,14,17,16)(15,18,19,20)(22,26)(27,33)(28,34)(35,36);(5,6,8,9)(7,10,11,12)(13,15,17,19)(14,18,16,20)(21,22)(23,27)(24,28)(25,26)(29,35)(30,33)(31,36)(32,34);(1,3,4,2)(21,23,25,30)(22,27,26,33)(24,31,32,29)(28,36,34,35);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,20,19,18)(21,24,25,32)(22,28,26,34)(23,29,30,31)(27,35,33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgh,d^2=fi,e^2=gh,b^a=bf,c^a=cg,c^b=ch,e^d=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 425869);