#616931

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2538 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=g,d^2=f,e^2=fi,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, 524127);