#6381979

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,4)(5,6,8,7)(9,10,14,19,15,20,12,17)(11,21,22,16,25,30,24,32)(13,18,29,36,28,35,26,33)(23,34,37,31,38,40,27,39);(5,7,8,6)(9,11)(10,16)(12,22)(13,27)(14,24)(15,25)(17,30)(18,34)(19,21)(20,32)(23,26)(28,37)(29,38)(31,33)(35,40)(36,39);(2,4)(9,12,15,14)(10,17,20,19)(11,22,25,24)(13,26,28,29)(16,30,32,21)(18,33,35,36)(23,37,38,27)(31,40,39,34);(9,13,12,26,15,28,14,29)(10,18,17,33,20,35,19,36)(11,23,24,27,25,38,22,37)(16,31,21,34,32,39,30,40);(13,28)(18,35)(23,38)(26,29)(27,37)(31,39)(33,36)(34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=i,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289175);