#772777

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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

2020 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)(6,9)(10,12)(14,17)(15,16)(21,22,25,24)(23,26,28,27);(13,14,18,17)(15,19,16,20)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(13,15,18,16)(14,19,17,20)(21,23)(22,26)(24,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,19,17,20)(21,22,25,24)(23,26,28,27);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,19,16,20)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fh,c^2=f,d^2=fh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=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, 679973);