#7007155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2027 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)(13,14,20,33,19,24,18,25)(15,26,17,21,29,39,28,23)(16,22,34,43,31,38,32,37)(27,40,30,35,41,44,42,36);(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,21,24,23)(16,27,31,41)(17,20,28,18)(22,35,38,36)(25,39,33,26)(30,34,42,32)(37,44,43,40);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,31)(14,22,24,38)(15,27,29,41)(17,30,28,42)(18,32,20,34)(21,35,23,36)(25,37,33,43)(26,40,39,44);(7,12)(9,11)(13,17,18,15,19,28,20,29)(14,23,25,39,24,21,33,26)(16,30,32,27,31,42,34,41)(22,36,37,44,38,35,43,40);(6,10)(9,11)(14,24)(16,31)(21,23)(25,33)(26,39)(27,41)(30,42)(32,34)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6914351);