#7203076

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,5)(4,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,30,34)(27,35,38,48)(28,32,39,46)(29,41,43,33)(31,42,45,37)(36,49,50,56)(40,53,52,47)(44,54,55,51);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22);(3,7)(5,8)(25,27,30,38)(26,31,34,45)(28,36,39,50)(29,42,43,37)(32,44,46,55)(33,48,41,35)(40,54,52,51)(47,56,53,49);(2,6)(3,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28,30,39)(26,32,34,46)(27,36,38,50)(29,40,43,52)(31,44,45,55)(33,47,41,53)(35,49,48,56)(37,51,42,54);(3,7)(5,8)(10,16)(14,17)(15,24)(22,23)(28,39)(32,46)(36,50)(40,52)(44,55)(47,53)(49,56)(51,54)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7110272);