#561125

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

1846 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 468321);