#157184

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

2624 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=eg,d^2=e,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, 64380);