#528265

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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

2438 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fgi,d^2=g,e^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 435461);