#557114

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

1886 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)(6,10)(8,12)(21,22,25,26)(23,29,31,27)(24,32,33,28)(30,34,36,35);(6,10)(7,11)(13,14,16,18)(15,19,20,17)(22,26)(24,33)(27,29)(30,36);(5,6,9,10)(7,8,11,12)(13,15,16,20)(14,17,18,19)(21,22)(23,27)(24,28)(25,26)(29,31)(30,34)(32,33)(35,36);(5,7,9,11)(6,8,10,12)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,35)(33,36);(5,8)(6,11)(7,10)(9,12)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dfg,e^b=ef

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, 464310);