#544589

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

1790 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)(8,10)(9,12)(22,26)(27,32)(28,33)(34,36);(5,6,8,10)(7,9,12,11)(13,14,16,18)(15,19,20,17)(23,29)(27,32)(30,35)(34,36);(5,6,8,10)(7,9,12,11)(13,15,16,20)(14,17,18,19)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36);(5,6)(7,9)(8,10)(11,12)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,7,8,12)(6,9,10,11)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=gh,d^2=fg,e^2=g,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, 451785);