#6483469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470884
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3179	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2053 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)(5,6)(7,12)(8,10)(9,11)(13,14,19,24)(15,21,27,36)(16,28,26,20)(17,22,31,37)(18,33,29,23)(25,35,39,44)(30,41,40,34)(32,43,42,38);(1,2,3,4)(7,8)(10,12)(13,15,18,26,19,27,29,16)(14,20,23,21,24,28,33,36)(17,25,32,40,31,39,42,30)(22,34,38,35,37,41,43,44);(5,7,9,8)(6,10,11,12)(13,16,29,27,19,26,18,15)(14,21,33,20,24,36,23,28)(17,30,42,39,31,40,32,25)(22,35,43,34,37,44,38,41);(5,8,9,7)(6,10,11,12)(13,17,19,31)(14,22,24,37)(15,25,27,39)(16,30,26,40)(18,32,29,42)(20,34,28,41)(21,35,36,44)(23,38,33,43);(6,11)(10,12)(17,31)(22,37)(25,39)(30,40)(32,42)(34,41)(35,44)(38,43)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6390665);