#7331519

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324820
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

1298 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7238715);