#6790649

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1223 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)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,29,31)(27,35,36,51)(28,39,40,33)(30,44,43,34)(32,46,45,38)(37,52,41,50)(42,49,53,55)(47,54,48,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(26,31)(27,36)(28,40)(34,44)(38,46)(42,53)(47,48)(50,52);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,30,38,29,36,43,46)(26,32,34,35,31,45,44,51)(28,41,42,56,40,37,53,54)(33,48,49,52,39,47,55,50);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,28)(26,33)(27,37)(29,40)(30,42)(31,39)(32,47)(34,49)(35,50)(36,41)(38,54)(43,53)(44,55)(45,48)(46,56)(51,52);(2,5)(6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,36)(28,40)(32,45)(33,39)(35,51)(38,46)(42,53)(49,55)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6697845);