#6379748

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2623 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6286944);