#7205239

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,d^2=f,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, 7112435);