#7053300

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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

1220 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fg,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6960496);