#158527

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
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

2516 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=ef,c^2=fg,d^2=h,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, 65723);