#6388679

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 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)(21,22,27,29,26,30,25,36)(23,31,32,34,33,24,28,35);(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23,27,32,26,33,25,28)(22,24,29,35,30,31,36,34);(2,4)(21,24)(22,28)(23,29)(25,34)(26,31)(27,35)(30,32)(33,36);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,25,26,27)(22,29,30,36)(23,32,33,28)(24,34,31,35);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(21,25,26,27)(22,29,30,36)(23,28,33,32)(24,35,31,34)

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,d^a=di,e^a=ef,d^b=df,e^c=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, 6295875);