#612696

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2906 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)(13,14,19,17)(15,20,18,16)(22,26)(23,24);(5,6)(7,10)(8,9)(11,12)(13,15,19,18)(14,16,17,20)(21,22,25,26)(23,27,24,28);(13,16)(14,18)(15,17)(19,20)(21,22)(23,27)(24,28)(25,26);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,17,19,14)(15,20,18,16)(21,23,25,24)(22,27,26,28);(5,7,8,11)(6,10,9,12)(13,18,19,15)(14,20,17,16)(21,24,25,23)(22,28,26,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519892);