#6381497

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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3167 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

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,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, 6288693);