#383038

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

730 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=h,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,g^c=gi

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, 290234);