#6565775

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24626	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1995 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)(7,9)(8,10)(11,12)(13,14,18,17)(15,19,21,23)(16,20,24,25)(22,26,27,28);(1,2,3,4)(6,9)(10,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(6,10)(8,12)(9,11)(15,21)(19,23)(22,27)(26,28);(6,9)(10,11)(13,16,18,24)(14,20,17,25)(15,22,21,27)(19,26,23,28);(13,17,18,14)(15,23,21,19)(16,20,24,25)(22,26,27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6472971);