#607217

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
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

1694 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)(7,8)(10,11)(15,19)(17,20)(21,22,26,24)(23,29,30,27)(25,28,33,32)(31,35,36,34);(5,6,9,12)(7,10,8,11)(13,14,16,18)(15,17,19,20)(22,24)(23,30)(25,33)(34,35);(1,3,4,2)(5,7,9,8)(6,10,12,11)(13,14)(15,17)(16,18)(19,20)(21,23,26,30)(22,27,24,29)(25,31,33,36)(28,34,32,35);(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,19)(18,20)(21,24,26,22)(23,29,30,27)(25,32,33,28)(31,35,36,34);(5,8,9,7)(6,11,12,10)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 514413);