#6395465

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6302661);