#7184823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092019);