#6466550

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373746);