#378148

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=i,d^2=efhi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285344);