#6597047

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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 ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2260 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)(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,21)(18,22)(19,20)(23,26)(24,28)(25,27);(5,7)(6,9)(8,12)(10,11)(13,15,17,14,19,22,18,20)(16,23,26,21,25,28,24,27);(2,4)(7,12)(9,10)(14,20)(15,22)(21,27)(23,28);(5,7)(6,10)(8,12)(9,11)(13,16,19,25)(14,21,20,27)(15,23,22,28)(17,26,18,24);(7,12)(9,10)(13,17,19,18)(14,15,20,22)(16,24,25,26)(21,28,27,23)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504243);