#6558194

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2641 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)(13,14,19,23)(15,24,25,28)(16,21,20,27)(17,22,18,26);(5,6)(7,10)(8,9)(11,12)(13,15,18,21,19,25,17,27)(14,20,22,24,23,16,26,28);(1,2,3,4)(6,9)(10,12)(15,25)(16,20)(21,27)(24,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,21)(15,26)(17,24)(18,28)(19,20)(22,25)(23,27);(7,11)(10,12)(13,17,19,18)(14,22,23,26)(15,27,25,21)(16,28,20,24)

Polycyclic presentation:

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

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, 6465390);