#7492737

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1212 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)(25,26,29,35,30,34,42,32)(27,31,37,46,38,45,53,43)(28,39,41,47,40,33,48,49)(36,50,52,54,51,44,55,56);(2,6)(3,4)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27,30,38)(26,31,34,45)(28,36,40,51)(29,37,42,53)(32,43,35,46)(33,44,39,50)(41,52,48,55)(47,54,49,56);(1,2)(3,8)(4,7)(5,6)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(29,42)(31,43)(33,47)(34,35)(37,53)(39,49)(41,48)(44,54)(45,46)(50,56)(52,55);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,30,40)(26,33,34,39)(27,36,38,51)(29,41,42,48)(31,44,45,50)(32,47,35,49)(37,52,53,55)(43,54,46,56);(1,4,5,3)(2,7,6,8)(10,18)(15,19)(16,17)(23,24)(26,34)(28,40)(31,45)(32,35)(36,51)(41,48)(43,46)(52,55)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=f,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,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, 7399933);