#6499803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6406999);