#441417

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,13,15,16)(17,18,22,25)(19,29,32,44)(20,33,36,43)(21,28,23,27)(24,38,39,31)(26,35,41,37)(30,40,46,48)(34,45,47,42);(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,34)(21,38)(22,32)(23,31)(25,39)(26,42)(27,44)(28,29)(30,35)(33,48)(36,47)(37,46)(40,43)(41,45);(1,3,4,8)(2,5,6,7)(11,15)(13,16)(18,25)(24,39)(26,41)(27,28)(29,44)(33,43)(40,48)(42,45);(17,20,22,36)(18,26,25,41)(19,30,32,46)(21,37,23,35)(24,40,39,48)(27,33,28,43)(29,45,44,42)(31,47,38,34)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 348613);