#6470195

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×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4259 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)(14,20)(15,22)(17,18)(21,27)(23,28)(25,26);(2,4)(5,7)(6,9)(8,11)(10,12)(13,14)(15,17)(16,21)(18,20)(19,22)(23,25)(24,28)(26,27);(7,11)(9,12)(13,15,19,20)(14,17,22,18)(16,23,24,27)(21,25,28,26);(7,11)(9,12)(13,16,19,24)(14,21,22,28)(15,23,20,27)(17,25,18,26);(6,10)(9,12)(16,24)(21,28)(23,27)(25,26)

Polycyclic presentation:

2^9|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,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, 6377391);