#836208

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2662 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)(6,9)(10,12)(13,14)(15,18)(16,19)(17,20)(22,27)(24,28);(13,14,17,20)(15,18,19,16)(21,22,26,27)(23,28,25,24);(1,2,3,4)(13,15)(14,16)(17,19)(18,20)(21,23,26,25)(22,28,27,24);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,24)(22,23)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,19,20,15)(21,25,26,23)(22,28,27,24)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 743404);