#6584161

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2170 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491357);