#585046

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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

1774 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)(5,6)(7,11)(8,12)(9,10)(13,14,17,20)(15,19,16,18)(21,22,25,28)(23,26,29,32)(24,27,31,34)(30,33,35,36);(1,2,4,3)(6,10)(7,12)(13,15,17,16)(14,18,20,19)(23,29)(26,32)(30,35)(33,36);(6,10)(7,12)(15,16)(18,19)(21,23,25,29)(22,26,28,32)(24,30,31,35)(27,33,34,36);(5,7)(6,8)(9,12)(10,11)(13,16,17,15)(14,19,20,18)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36);(1,3,4,2)(5,8)(6,7)(9,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,24,25,31)(22,27,28,34)(23,30,29,35)(26,33,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=fghi,c^2=fgh,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df

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, 492242);