#6520558

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2429 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)(13,14,20,25)(15,17,23,22)(16,21,26,28)(18,24,19,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,16,20,23,18,26)(14,21,24,17,25,28,27,22);(6,9)(7,11)(13,16,18,15,20,26,19,23)(14,22,27,28,25,17,24,21);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,25)(16,27)(18,21)(19,28)(20,22)(24,26);(6,9)(10,12)(13,18,20,19)(14,24,25,27)(15,26,23,16)(17,28,22,21)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,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, 6427754);