#6470212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

4235 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)(7,9)(8,10)(11,12)(13,14,20,17)(15,25,22,28)(16,26,21,27)(18,24,19,23);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,21)(16,17)(18,27)(19,26)(20,22)(23,28)(24,25);(7,11)(9,12)(13,16,20,21)(14,22,17,15)(18,25,19,28)(23,26,24,27);(7,11)(9,12)(13,17,20,14)(15,21,22,16)(18,24,19,23)(25,27,28,26);(6,10)(9,12)(14,17)(15,22)(23,24)(26,27)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,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^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=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, 6377408);