#6535912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2521 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6443108);