#160410

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(3,7)(5,8)(11,15)(14,16)(17,18,21,19)(20,23,24,22)(25,26)(27,31)(28,29)(30,32)(34,35)(36,40);(10,13)(14,16)(20,24)(22,23)(26,29)(27,32)(33,34,37,35)(36,38,40,39);(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22)(25,26)(27,31)(28,29)(30,32)(33,35,37,34)(36,38,40,39);(1,3)(2,5)(4,7)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,30)(28,32)(29,31)(33,36,37,40)(34,38,35,39)

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=efh,d^2=eg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

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, 67606);