#7238280

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1329 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)(25,26,30,35)(27,37,40,32)(28,34,42,52)(29,36,31,33)(38,50,41,47)(39,48,44,46)(43,49,45,51)(53,55,54,56);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,32)(28,41)(29,44)(30,40)(31,39)(33,46)(34,50)(35,37)(36,48)(38,42)(43,53)(45,54)(47,52)(49,56)(51,55);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,33)(28,42)(29,31)(32,46)(34,51)(35,36)(37,48)(38,41)(39,44)(47,56)(49,52)(50,55);(3,8)(5,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,42)(26,34,35,52)(27,38,40,41)(29,45,31,43)(32,47,37,50)(33,49,36,51)(39,54,44,53)(46,55,48,56);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(25,29,30,31)(26,33,35,36)(27,39,40,44)(28,43,42,45)(32,48,37,46)(34,51,52,49)(38,53,41,54)(47,55,50,56)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7145476);