#6776961

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1251 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6684157);