#544387

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

1902 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=f,e^2=g,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, 451583);