#6470802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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

2150 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:

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

Polycyclic presentation:

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

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, 6377998);