#191150

order	512
nilpotency class	3
isoclinism	isoclinic class of #188766
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,g^b=gh,g^c=gh

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, 98346);