#7205497

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

1352 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)(25,26,30,32)(27,37,38,50)(28,41,42,34)(29,36,31,35)(33,45,46,40)(39,53,52,51)(43,48,44,49)(47,55,56,54);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(26,32)(27,38)(28,42)(35,36)(40,45)(43,44)(47,56)(51,53);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,30,38)(26,33,32,46)(28,39,42,52)(29,40,31,45)(34,47,41,56)(35,37,36,50)(43,54,44,55)(48,53,49,51);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,39)(29,44)(30,42)(31,43)(32,41)(33,47)(35,49)(36,48)(37,51)(38,52)(40,55)(45,54)(46,56)(50,53);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(25,29,30,31)(26,35,32,36)(27,40,38,45)(28,43,42,44)(33,37,46,50)(34,48,41,49)(39,54,52,55)(47,53,56,51)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,d^2=fi,e^2=i,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=ehi,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, 7112693);