#6526765

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1836 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)(21,22,27,31)(23,29,24,33)(25,30,26,35)(28,34,36,32);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,27,24)(22,28,31,36)(25,32,26,34)(29,30,33,35);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(21,24,27,23)(22,29,31,33)(25,32,26,34)(28,30,36,35);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(21,25,27,26)(22,30,31,35)(23,32,24,34)(28,33,36,29);(7,13)(10,16)(14,19)(17,20)(21,25,27,26)(22,30,31,35)(23,32,24,34)(28,33,36,29)

Polycyclic presentation:

2^9|a^2=hi,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^c=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, 6433961);