#6388843

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1779 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)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,28,23,27,32,26,31)(24,29,34,33,35,36,25,30);(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23,28,32,27,31,26,22)(24,33,34,36,35,30,25,29);(2,4)(21,24)(22,29)(23,33)(25,26)(27,35)(28,34)(30,31)(32,36);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,25,27,34)(22,30,32,33)(23,29,31,36)(24,26,35,28);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(21,26,27,28)(22,31,32,23)(24,34,35,25)(29,33,36,30)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=dfi,e^c=ei,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, 6296039);