#6526801

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1799 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,29)(24,28)(25,27)(26,30)(31,35)(32,34)(33,36);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(22,27)(23,25)(28,34)(29,30)(31,32)(35,36);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(21,23,26,25)(22,27,30,29)(24,31,33,32)(28,34,36,35);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,24,26,33)(22,28,30,36)(23,31,25,32)(27,34,29,35);(7,13)(10,16)(14,19)(17,20)(21,23,26,25)(22,29,30,27)(24,32,33,31)(28,34,36,35)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,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, 6433997);