#7203776

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1394 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,34)(27,35,36,48)(28,39,40,32)(29,42,44,33)(31,43,45,38)(37,51,50,49)(41,47,55,54)(46,52,53,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(27,36)(28,40)(31,45)(32,39)(35,48)(38,43)(41,55)(47,54);(3,7)(5,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27)(26,31)(28,37)(29,43)(30,36)(32,46)(33,48)(34,45)(35,42)(38,44)(39,53)(40,50)(41,56)(47,51)(49,54)(52,55);(2,6)(3,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,40)(26,32,34,39)(27,37,36,50)(29,41,44,55)(31,46,45,53)(33,47,42,54)(35,49,48,51)(38,52,43,56);(3,7)(5,8)(10,18)(15,19)(16,17)(23,24)(27,36)(28,40)(31,45)(32,39)(35,48)(38,43)(41,55)(47,54)

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=fi,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^c=ei,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, 7110972);