#6376393

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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)(25,33,35,36,34,29,32,31);(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23,27,30,26,28,24,22)(25,31,35,33,34,36,32,29);(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)(25,32,34,35)(29,36,33,31);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(21,25,26,34)(22,29,30,33)(23,31,28,36)(24,32,27,35);(25,34)(29,33)(31,36)(32,35)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,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, 6283589);