#157445

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×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

2708 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

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, 64641);