#217067

order	512
nilpotency class	3
isoclinism	isoclinic class of #162948
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=eh,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,f^c=fi,g^c=ghi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124263);