#814254

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

2646 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=i,e^2=fg,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, 721450);