#378146

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1465 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)(9,10,14,19,15,17,13,20)(11,21,22,16,24,31,25,32)(12,26,27,18,28,34,29,33)(23,30,36,39,38,40,37,35);(1,2,4,3)(5,6)(7,8)(9,11,13,25,15,24,14,22)(10,16,20,21,17,32,19,31)(12,23,29,37,28,38,27,36)(18,30,26,35,33,40,34,39);(6,8)(10,17)(11,22)(13,14)(16,21)(18,33)(23,36)(24,25)(27,29)(30,35)(31,32)(37,38)(39,40);(1,3,4,2)(9,12,15,28)(10,18,17,33)(11,23,24,38)(13,29,14,27)(16,30,32,40)(19,34,20,26)(21,35,31,39)(22,36,25,37)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,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, 285342);