#7492804

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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

1189 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7400000);