#850556

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 757752);