#200092

order	512
nilpotency class	3
isoclinism	isoclinic class of #199833
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1147 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)(3,7,11,16)(4,8)(5,12)(6,9)(10,14,13,15)(17,18,22,26,23,27,21,37)(19,28,32,39,30,43,31,24)(20,33,35,25,36,40,34,41)(29,38,46,42,44,48,45,47);(1,3)(2,7,9,16)(4,10)(5,13)(6,11)(8,14,12,15)(17,19)(18,24,27,39)(20,29)(21,31)(22,32)(23,30)(25,38,41,48)(26,28,37,43)(33,47,40,42)(34,45)(35,46)(36,44);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,14,16,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,36)(24,38)(26,40)(27,41)(28,42)(30,44)(31,45)(32,46)(33,37)(39,48)(43,47);(1,5,6,4)(2,8,9,12)(3,10,11,13)(7,15,16,14)(19,30)(24,39)(28,43)(29,44)(31,32)(38,48)(42,47)(45,46)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=i,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,f^c=fhi

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, 107288);