#6558052

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,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, 6465248);