#6954147

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1369 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fh,d^2=fhi,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=dfh,e^c=ei,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, 6861343);