#7205113

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 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)(27,37,40,50)(28,34,41,46)(29,36,31,35)(33,44,45,39)(38,51,52,56)(42,49,43,48)(47,53,55,54);(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)(28,41)(33,45)(35,36)(37,50)(38,52)(42,43)(53,54);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,30,40)(26,33,32,45)(28,38,41,52)(29,39,31,44)(34,47,46,55)(35,37,36,50)(42,53,43,54)(48,56,49,51);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,38)(29,43)(30,41)(31,42)(32,46)(33,47)(35,49)(36,48)(37,51)(39,54)(40,52)(44,53)(45,55)(50,56);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(25,29,30,31)(26,35,32,36)(27,39,40,44)(28,42,41,43)(33,37,45,50)(34,48,46,49)(38,53,52,54)(47,56,55,51)

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=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7112309);