#6483442

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2185 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=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, 6390638);