#613432

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

3090 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)(7,11)(10,12)(14,18)(15,16)(21,22,25,26)(23,28,27,24);(5,6,9,8)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(22,26)(23,27);(5,7)(6,10)(8,12)(9,11)(13,15,17,16)(14,19,18,20)(21,23,25,27)(22,24,26,28);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16,17,15)(14,20,18,19);(5,8,9,6)(7,12,11,10)(13,14)(15,19)(16,20)(17,18)(21,24)(22,27)(23,26)(25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,d^2=fh,e^2=fg,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, 520628);