#7199902

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1266 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=gi,d^2=fhi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7107098);