#544322

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

1982 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)(7,9)(8,10)(11,12)(21,22)(23,32)(24,28)(25,29)(26,30)(27,31)(33,36)(34,35);(5,7)(6,9)(8,11)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23,27,24)(22,28,31,32)(25,34,26,33)(29,36,30,35);(6,10)(9,12)(14,17)(18,20)(21,24,27,23)(22,28,31,32)(25,33,26,34)(29,36,30,35);(7,11)(9,12)(21,25,27,26)(22,29,31,30)(23,33,24,34)(28,35,32,36);(13,15,16,19)(14,18,17,20)(21,26,27,25)(22,30,31,29)(23,33,24,34)(28,35,32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=f,e^2=fh,b^a=bf,c^a=cg,c^b=ch,d^b=dfg,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, 451518);