#899991

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2022 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:

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

Polycyclic presentation:

2^9|b^2=fghi,c^2=fh,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 807187);