#6916103

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fgi,d^2=fi,e^2=f,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^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6823299);