#6487723

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2192 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)(7,10)(8,11)(9,12)(15,23)(16,17)(19,27)(20,21)(24,28)(25,26);(1,2,3,4)(7,8)(10,11)(13,14)(15,24)(16,20)(17,21)(18,22)(19,25)(23,28)(26,27);(5,7,9,8)(6,10,12,11)(13,15,18,25)(14,19,22,24)(16,23,17,26)(20,27,21,28);(5,8,9,7)(6,11,12,10)(13,16,18,17)(14,20,22,21)(15,23,25,26)(19,27,24,28);(6,12)(10,11)(13,16,18,17)(14,21,22,20)(15,23,25,26)(19,28,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394919);