#7149200

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	#42652	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1911 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)(13,14,19,21,18,24,17,23)(15,20,31,36,25,35,27,33)(16,22,32,37,29,40,30,39)(26,34,42,44,28,38,41,43);(1,2,3,4)(6,9)(7,11)(13,15)(14,20)(16,28)(17,27)(18,25)(19,31)(21,36)(22,38)(23,33)(24,35)(26,29)(30,42)(32,41)(34,40)(37,43)(39,44);(5,6)(7,10)(8,9)(11,12)(14,21)(15,25)(16,29)(17,19)(20,33)(22,39)(23,24)(34,44)(35,36)(37,40)(38,43)(41,42);(2,4)(6,9)(10,12)(13,16)(14,22)(15,26)(17,30)(18,29)(19,32)(20,34)(21,37)(23,39)(24,40)(25,28)(27,41)(31,42)(33,43)(35,38)(36,44);(5,7,8,11)(6,10,9,12)(15,25)(16,29)(20,35)(22,40)(27,31)(30,32)(33,36)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,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, 7056396);