#6526806

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1747 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)(23,33)(24,31)(25,30)(26,29)(27,32)(28,34)(35,36);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,27,25)(22,28,32,36)(24,34,26,35)(29,33,31,30);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(21,24,27,26)(22,29,32,31)(23,34,25,35)(28,33,36,30);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,25,27,23)(22,30,32,33)(24,34,26,35)(28,29,36,31);(7,13)(10,16)(14,19)(17,20)(21,24,27,26)(22,31,32,29)(23,34,25,35)(28,30,36,33)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=hi,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, 6434002);