#6381508

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3196 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)(11,19)(12,16)(13,26)(14,17)(15,22)(18,29)(20,37)(21,35)(23,40)(24,36)(25,33)(27,31)(28,34)(30,39)(32,38);(5,7,8,6)(9,11,14,21)(10,15,17,31)(12,23,25,39)(13,27,29,22)(16,32,33,37)(18,35,26,19)(20,28,38,24)(30,36,40,34);(2,4);(9,12,14,25)(10,16,17,33)(11,20,21,38)(13,28,29,24)(15,30,31,40)(18,36,26,34)(19,37,35,32)(22,39,27,23);(10,17)(11,21)(16,33)(18,26)(20,38)(22,27)(23,39)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,e^b=ei,f^b=fi,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, 6288704);