#921792

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1939 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)(5,6)(7,11)(8,9)(10,12)(13,14,18,17)(15,20,16,19)(21,22,26,24)(23,28,25,27);(6,9)(10,11)(13,14,18,17)(15,20,16,19)(21,23,26,25)(22,27,24,28);(6,9)(10,11)(13,15,18,16)(14,19,17,20)(21,23,26,25)(22,28,24,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,20,17,19)(21,24,26,22)(23,28,25,27);(5,7,8,12)(6,10,9,11)(13,17,18,14)(15,20,16,19)(21,25,26,23)(22,28,24,27)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 828988);