#1428394

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2628 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:

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

Polycyclic presentation:

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

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, 1335590);