#200384

order	512
nilpotency class	3
isoclinism	isoclinic class of #200158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1041 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)(3,11,4,7)(5,9)(6,10)(8,13,15,12)(14,16)(17,18,22,27,23,28,21,26)(19,29,31,39,33,42,32,24)(20,34,36,40,37,25,35,41)(30,43,44,38,46,48,45,47);(1,3,5,4)(2,7)(6,13,14,12)(8,10)(9,11)(15,16)(17,19)(18,24,28,39)(20,30)(21,32)(22,31)(23,33)(25,38,34,47)(26,42,27,29)(35,45)(36,44)(37,46)(40,43,41,48);(1,4,5,3)(2,8,9,15)(6,12,14,13)(7,10,11,16)(17,20,23,37)(18,25,28,34)(19,30,33,46)(21,36,22,35)(24,38,39,47)(26,41,27,40)(29,43,42,48)(31,45,32,44);(2,9)(3,4)(10,16)(12,13)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,42)(25,40,34,41)(30,44,46,45)(38,48,47,43)

Polycyclic presentation:

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

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, 107580);