#372707

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1457 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)(5,6)(7,8)(9,10)(11,21,26,34)(12,27)(13,30,22,36)(14,19)(15,20)(16,24,35,25)(17,23)(18,28,38,31)(29,39)(32,37)(33,40);(1,2,4,3)(5,6)(7,8)(9,11,14,25,15,26,33,24)(10,16,19,34,20,35,40,21)(12,28,29,13,23,31,32,22)(17,36,37,38,27,30,39,18);(6,8)(9,12,15,23)(10,17,20,27)(11,22,26,13)(14,32,33,29)(16,18,35,38)(19,39,40,37)(21,36,34,30)(24,28,25,31);(1,3,4,2)(9,13,15,22)(10,18,20,38)(11,23,26,12)(14,31,33,28)(16,17,35,27)(19,36,40,30)(21,39,34,37)(24,29,25,32)

Polycyclic presentation:

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

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, 279903);