#1259294

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

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

5433 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280903, #72281051, #72281166
maximal normal subgroup of	#72280903, #72281051, #72281166

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1166490);