#1465619

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

2351 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fgi,d^2=h,e^2=g,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, 1372815);