#557779

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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

1926 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)(6,9)(10,12)(13,14,17,16)(15,18,20,19)(21,22,26,27)(23,28,29,24)(25,30,33,34)(31,35,36,32);(7,11)(10,12)(13,14,17,16)(15,19,20,18)(22,27)(24,28)(25,33)(31,36);(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,18,16,19)(21,23,26,29)(22,28,27,24)(25,31,33,36)(30,35,34,32);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,16,17,14)(15,19,20,18)(21,24,26,28)(22,29,27,23)(25,32,33,35)(30,36,34,31);(5,6)(7,10)(8,9)(11,12)(21,25,26,33)(22,30,27,34)(23,31,29,36)(24,32,28,35)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fh,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

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, 464975);