#873661

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

1984 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)(6,11)(9,12)(13,14,17,19)(15,16,20,18);(2,3)(5,6)(7,9)(8,12)(10,11)(13,15,17,20)(14,18,19,16)(21,22,25,24)(23,27,28,26);(1,2,4,3)(5,7,10,8)(6,9,11,12)(13,16)(14,15)(17,18)(19,20)(21,23,25,28)(22,26,24,27);(1,3,4,2)(5,8,10,7)(6,12,11,9)(13,15)(14,16)(17,20)(18,19);(5,9,10,12)(6,7,11,8)(13,16)(14,15)(17,18)(19,20)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=fghi,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 780857);