#7182106

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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1767 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)(15,23)(16,28)(17,32)(18,21)(19,27)(20,30)(22,33)(24,40)(25,36)(26,35)(29,39)(31,38)(34,42)(37,41)(43,44);(5,6,9,10)(7,8,12,11)(13,15)(14,19)(16,29)(17,25)(18,26)(20,37)(21,36)(22,35)(23,32)(24,30)(27,33)(28,34)(31,41)(38,39)(40,43)(42,44);(2,4)(5,7)(6,11)(8,10)(9,12)(13,16)(14,20)(15,24)(17,31)(18,30)(19,34)(21,28)(22,38)(23,39)(25,42)(26,29)(27,41)(32,43)(33,44)(35,40)(36,37);(6,10)(7,12)(15,25)(16,30)(17,33)(19,35)(20,28)(22,32)(23,36)(24,41)(26,27)(29,42)(34,39)(37,40);(5,8,9,11)(6,7,10,12)(14,21)(15,26)(16,30)(22,32)(25,27)(31,44)(34,37)(39,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,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, 7089302);