#7193413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1662 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7100609);