#7182120

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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,e^2=hi,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, 7089316);