#545524

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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

2406 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)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,28)(31,32);(2,4)(9,10,12,13)(11,14,16,15)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32);(1,2,3,4)(5,7,8,6)(10,13)(14,15)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32);(9,10)(11,14)(12,13)(15,16)(17,20,22,23)(18,24,26,19)(21,28,30,31)(25,32,29,27);(9,11,12,16)(10,14,13,15)(17,21,22,30)(18,25,26,29)(19,27,24,32)(20,28,23,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=fg,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,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, 452720);