#7234279

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1166 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)(3,6)(4,7)(5,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,35)(28,40)(29,31)(30,43)(32,39)(33,41)(34,37)(36,48)(38,52)(42,46)(44,50)(45,56)(47,49)(51,55)(53,54);(3,4)(6,7)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(26,31)(27,36)(33,40)(34,43)(38,49)(39,50)(46,55)(53,56);(1,3,5,4)(2,6,8,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,37,50,29,36,30,39)(26,32,43,48,31,44,34,35)(28,38,51,56,41,49,42,53)(33,45,55,52,40,54,46,47);(1,3,5,4)(2,7,8,6)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,38)(29,41)(30,42)(31,40)(32,45)(34,46)(35,47)(36,49)(37,51)(39,53)(43,55)(44,54)(48,52)(50,56);(1,4,5,3)(2,7,8,6)(10,18)(15,19)(16,17)(23,24)(26,31)(28,41)(32,44)(34,43)(35,48)(38,49)(42,51)(53,56)

Polycyclic presentation:

2^9|b^2=fi,c^2=fghi,d^2=fh,e^2=hi,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=ehi,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, 7141475);