#6496178

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470092
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3482	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3048 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,31)(23,32,35,36)(24,29,25,30)(27,33,28,34);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,35)(22,27,31,28)(24,33,25,34)(29,32,30,36);(7,13)(10,17)(14,19)(18,20)(21,23)(22,28)(24,33)(25,34)(26,35)(27,31)(29,36)(30,32);(7,13)(10,17)(14,19)(18,20)(21,24,26,25)(22,29,31,30)(23,33,35,34)(27,32,28,36);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,24,26,25)(22,30,31,29)(23,33,35,34)(27,36,28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,f^b=fi,g^b=gi,f^c=fi,g^c=gi

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, 6403374);