#6685784

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(22,28)(23,33)(24,25)(32,35);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,22,26,28)(23,31,33,36)(24,30,25,29)(27,35,34,32);(1,2,3,4)(21,23)(22,29)(24,27)(25,34)(26,33)(28,30)(31,35)(32,36);(6,11)(10,13)(12,19)(18,20)(21,24,26,25)(22,30,28,29)(23,34,33,27)(31,32,36,35);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,25,26,24)(22,31,28,36)(23,34,33,27)(29,35,30,32)

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592980);