#7205233

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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)(27,36)(28,40)(29,34)(30,44)(31,35)(32,39)(33,41)(37,48)(38,53)(42,50)(43,49)(45,46)(47,54)(51,52)(55,56);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(27,37)(28,41)(32,46)(33,40)(36,48)(39,45)(42,43)(49,50);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27)(26,32)(28,38)(29,39)(30,45)(31,37)(33,47)(34,48)(35,46)(36,44)(40,56)(41,52)(42,54)(43,55)(49,51)(50,53);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,31,41)(26,33,35,40)(27,38,37,52)(29,43,30,42)(32,47,46,56)(34,50,44,49)(36,51,48,53)(39,55,45,54);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(25,29,31,30)(26,34,35,44)(27,39,37,45)(28,42,41,43)(32,48,46,36)(33,49,40,50)(38,54,52,55)(47,51,56,53)

Polycyclic presentation:

2^9|a^2=f,b^2=f,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=ehi,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, 7112429);