#6470724

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2219 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

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, 6377920);