#874487

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

1912 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)(6,9)(7,11)(13,14,18,17)(15,20,16,19)(21,22,25,27)(23,24,26,28);(5,6)(7,10)(8,9)(11,12)(13,14,18,17)(15,20,16,19)(21,23)(22,24)(25,26)(27,28);(13,15,18,16)(14,19,17,20)(21,24)(22,26)(23,27)(25,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,20,17,19);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,20,16,19)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=fi,e^2=fg,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, 781683);