#7149140

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	#45101	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1886 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)(13,14,19,33,18,23,17,22)(15,24,26,20,28,35,27,36)(16,29,30,38,32,21,31,37)(25,40,43,44,42,39,41,34);(1,2,3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,15,18,28)(14,20,23,36)(16,25,32,42)(17,26,19,27)(21,34,29,44)(22,35,33,24)(30,41,31,43)(37,39,38,40);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,31)(18,32)(19,30)(20,34)(22,38)(23,29)(24,39)(26,43)(27,41)(28,42)(33,37)(35,40)(36,44);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,33)(15,26,28,27)(16,30,32,31)(20,35,36,24)(21,37,29,38)(25,41,42,43)(34,39,44,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,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, 7056336);