#545722

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, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1872 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)(7,11)(8,9)(10,12)(21,22,26,23)(24,27,32,29)(25,30,33,28)(31,35,36,34);(5,6,8,9)(7,10,12,11)(14,17)(18,20)(22,23)(27,29)(28,30)(34,35);(1,3,4,2)(6,9)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35);(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,18,17,20)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36);(5,7)(6,10)(8,12)(9,11)(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,d^2=fh,e^2=fg,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, 452918);