#7003184

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	#42908	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,25)(16,22)(17,23)(18,34)(19,24)(20,29)(21,32)(26,42)(27,35)(28,37)(30,41)(31,39)(33,38)(36,44)(40,43);(5,6)(7,11)(8,10)(9,12)(13,15,17,28,19,30,18,29)(14,20,34,41,24,37,23,25)(16,26,33,27,32,44,31,43)(21,35,38,42,22,40,39,36);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,33)(18,31)(19,32)(20,35)(22,24)(23,39)(25,36)(27,28)(29,43)(30,44)(34,38)(37,40)(41,42);(7,12)(9,11)(13,16)(14,22)(15,27)(17,31)(18,33)(19,32)(20,36)(21,24)(23,39)(25,35)(26,28)(29,44)(30,43)(34,38)(37,42)(40,41);(6,10)(9,11)(13,17,19,18)(14,23,24,34)(15,28,30,29)(16,31,32,33)(20,25,37,41)(21,38,22,39)(26,43,44,27)(35,42,40,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 6910380);