#759489

order	512
nilpotency class	2
isoclinism	isoclinic class of #759288
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

3022 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)(5,6)(7,11)(8,12)(9,10)(14,15)(16,20)(21,22,26,28)(23,24,25,27);(6,10)(7,12)(13,14,17,15)(16,18,20,19)(21,23,26,25)(22,27,28,24);(5,7,9,12)(6,11,10,8)(13,15,17,14)(16,18,20,19)(21,24,26,27)(22,23,28,25);(1,2)(3,4);(5,8,9,11)(6,7,10,12)(13,16,17,20)(14,18,15,19)(21,25,26,23)(22,24,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 666685);