#414060

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

956 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)(17,18,21,24)(19,29,31,42)(20,33,34,41)(22,28,23,27)(25,37,38,32)(26,35,39,36)(30,44,45,40)(43,47,48,46);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(18,24)(25,38)(26,39)(27,28)(29,42)(33,41)(40,44)(43,48);(10,13)(14,16)(17,19)(18,25)(20,30)(21,31)(22,37)(23,32)(24,38)(26,40)(27,42)(28,29)(33,43)(34,45)(35,47)(36,46)(39,44)(41,48);(9,11,12,15)(10,14,13,16)(17,20,23,36,21,34,22,35)(18,26,28,33,24,39,27,41)(19,30,32,46,31,45,37,47)(25,40,29,43,38,44,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321256);