#6443838

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2085 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)(13,14,18,21)(15,19,25,35)(16,20,28,38)(17,22,31,39)(23,32,41,43)(24,33,26,34)(27,36,42,44)(29,37,30,40);(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,27)(17,24)(18,25)(20,36)(21,35)(22,33)(23,29)(26,31)(28,42)(30,41)(32,37)(34,39)(38,44)(40,43);(1,2,3,4)(13,16)(14,20)(15,23)(17,30)(18,28)(19,32)(21,38)(22,40)(24,27)(25,41)(26,42)(29,31)(33,36)(34,44)(35,43)(37,39);(5,8,9,7)(6,10,11,12)(15,24)(17,31)(19,33)(22,39)(23,27)(25,26)(29,30)(32,36)(34,35)(37,40)(41,42)(43,44);(14,21)(15,25)(16,28)(22,39)(24,26)(29,30)(32,43)(36,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351034);