#158495

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

2508 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)(13,14,17,18)(15,19,20,16)(21,22)(23,27)(24,26)(25,28);(2,4)(10,12)(15,20)(16,19)(21,22,24,26)(23,27,28,25);(5,6,8,7)(9,10,11,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7,8,6)(13,16)(14,15)(17,19)(18,20)(21,23)(22,25)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=efh,b^2=e,c^2=efg,d^2=eh,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, 65691);