#6481917

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1786 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6389113);