#6490873

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3377 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)(7,11)(10,12)(13,14)(15,20)(16,21)(17,22)(18,23)(19,25)(24,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,23,27)(16,25,17,24)(20,21,28,22);(2,4)(6,9)(10,12)(13,15)(14,20)(16,24)(17,25)(18,26)(19,22)(21,27)(23,28);(6,9)(10,12)(13,16,18,17)(14,21,23,22)(15,24,26,25)(19,20,27,28);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,22,23,21)(15,25,26,24)(19,20,27,28)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6398069);