#6437227

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433040
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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2236 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)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22)(23,29)(24,32)(25,27)(26,31)(28,33)(30,35)(34,36);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,24,31,25,30,33,36)(22,26,28,29,27,34,32,35);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(23,30)(26,34)(29,35)(31,36);(6,11)(10,16)(12,18)(17,20)(22,27)(23,30)(28,32)(31,36);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(23,30)(26,34)(29,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344423);