#198532

order	512
nilpotency class	3
isoclinism	isoclinic class of #198528
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60946	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1076 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105728);