#7149188

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	#42880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1904 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7056384);