#6389621

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34880	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1796 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,27,23,26,30,24,28)(25,29,33,31,34,36,32,35);(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,14)(18,20)(21,23,27,30,26,28,24,22)(25,31,33,36,34,35,32,29);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,26,27)(22,28,30,23)(25,32,34,33)(29,35,36,31);(7,16)(11,14)(15,17)(19,20)(21,25)(22,29)(23,31)(24,32)(26,34)(27,33)(28,35)(30,36);(5,9,10,8)(6,12,13,18)(7,17,16,15)(11,19,14,20)(21,24,26,27)(22,28,30,23)(25,33,34,32)(29,31,36,35)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=dfi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296817);