#7205450

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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

1350 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)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,35)(28,33)(29,31)(30,42)(32,38)(34,41)(36,49)(37,48)(39,45)(40,53)(43,44)(46,51)(47,52)(50,56)(54,55);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(26,31)(32,44)(33,45)(34,42)(35,48)(46,55)(47,53)(49,56);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27)(26,32)(28,36)(29,37)(30,43)(31,44)(33,46)(34,48)(35,42)(38,41)(39,50)(40,54)(45,55)(47,56)(49,53)(51,52);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,39)(26,33,31,45)(27,36,37,50)(30,40,41,52)(32,46,44,55)(34,47,42,53)(35,49,48,56)(38,51,43,54);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(26,31)(27,37)(28,39)(34,42)(38,43)(40,52)(46,55)(49,56)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7112646);