#691943

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 599139);