#6915173

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1389 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)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26,30,35)(27,31,38,34)(28,39,41,43)(29,33,42,48)(32,45,47,36)(37,44,51,49)(40,52,53,54)(46,55,56,50);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,24)(18,19)(25,27,30,38)(26,31,35,34)(28,36,41,45)(29,37,42,51)(32,43,47,39)(33,44,48,49)(40,50,53,55)(46,54,56,52);(1,3)(2,5)(4,8)(6,7)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28,38,45,30,41,27,36)(26,32,31,43,35,47,34,39)(29,40,51,55,42,53,37,50)(33,46,44,54,48,56,49,52);(3,8)(5,7)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(25,29)(26,33)(27,37)(28,40)(30,42)(31,44)(32,46)(34,49)(35,48)(36,50)(38,51)(39,52)(41,53)(43,54)(45,55)(47,56);(2,6)(5,7)(9,14,15,13)(10,17,18,12)(11,21,22,23)(16,20,19,24)(29,42)(33,48)(37,51)(40,53)(44,49)(46,56)(50,55)(52,54)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6822369);