#612970

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

2974 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)(15,16,20,18)(22,24)(23,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,17)(19,20)(21,22,25,24)(23,28,27,26);(13,16,19,18)(14,15,17,20)(21,23,25,27)(22,26,24,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,17,19,14)(15,16,20,18)(21,22,25,24)(23,26,27,28);(5,7,8,11)(6,10,9,12)(13,18,19,16)(14,20,17,15)(21,24,25,22)(23,28,27,26)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^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, 520166);