#592494

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1908 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fghi,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,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, 499690);