#557388

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=hi,d^2=g,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, 464584);