#199405

order	512
nilpotency class	3
isoclinism	isoclinic class of #199242
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1089 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:

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

Polycyclic presentation:

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

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, 106601);