#7182088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1740 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)(15,26)(16,22)(17,23)(18,24)(19,34)(20,25)(21,29)(27,37)(28,36)(30,38)(31,40)(32,44)(33,41)(35,39)(42,43);(5,6,9,10)(7,8,12,11)(13,15)(14,21)(16,27)(17,32)(18,29)(19,35)(20,30)(22,36)(23,42)(24,38)(25,39)(26,34)(28,33)(31,43)(37,40)(41,44);(2,4)(5,7)(6,11)(8,10)(9,12)(13,16,20,17)(14,22,25,23)(15,27,30,32)(18,33,19,31)(21,36,39,42)(24,41,34,40)(26,37,38,44)(28,35,43,29);(6,10)(7,12)(13,17,20,16)(14,23,25,22)(15,28,30,43)(18,33,19,31)(21,37,39,44)(24,41,34,40)(26,36,38,42)(27,35,32,29);(5,8,9,11)(6,7,10,12)(13,18,20,19)(14,24,25,34)(15,29,30,35)(16,31,17,33)(21,38,39,26)(22,40,23,41)(27,43,32,28)(36,37,42,44)

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,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089284);