#7187368

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1191 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)(11,16)(12,17)(13,21)(14,15)(18,22)(19,23)(20,24)(25,26,30,32)(27,36,39,48)(28,34,40,46)(29,35,31,44)(33,42,45,38)(37,51,52,56)(41,49,43,50)(47,53,55,54);(1,2,5,6)(3,7,4,8)(10,15)(11,18)(12,13)(23,24)(26,32)(28,40)(33,45)(35,44)(36,48)(37,52)(41,43)(53,54);(1,3,5,4)(2,7,6,8)(9,11,14,18)(10,16,15,22)(12,20,13,19)(17,24,21,23)(25,27,29,42,30,39,31,38)(26,33,44,48,32,45,35,36)(28,37,43,53,40,52,41,54)(34,47,49,56,46,55,50,51);(1,4,5,3)(2,7,6,8)(9,12,14,13)(10,17,15,21)(11,19,18,20)(16,23,22,24)(25,28)(26,34)(27,37)(29,43)(30,40)(31,41)(32,46)(33,47)(35,50)(36,51)(38,54)(39,52)(42,53)(44,49)(45,55)(48,56);(9,13,14,12)(10,17,15,21)(11,19,18,20)(16,24,22,23)(25,29,30,31)(26,35,32,44)(27,38,39,42)(28,41,40,43)(33,48,45,36)(34,49,46,50)(37,53,52,54)(47,51,55,56)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fg,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7094564);