#199416

order	512
nilpotency class	3
isoclinism	isoclinic class of #199251
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

1088 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18,22,27,23,26,21,34)(19,24,30,39,29,40,36,38)(20,31,33,25,32,42,35,41)(28,43,46,47,44,48,45,37);(1,3,4,13,6,12,5,11)(2,7,14,15,9,16,8,10)(17,19,21,30,23,29,22,36)(18,24,27,38,26,40,34,39)(20,28,35,46,32,44,33,45)(25,37,42,48,41,47,31,43);(1,2)(3,7)(4,14)(5,8)(6,9)(10,11)(12,16)(13,15)(17,20)(18,25)(19,28)(21,35)(22,33)(23,32)(24,37)(26,41)(27,42)(29,44)(30,46)(31,34)(36,45)(38,48)(39,43)(40,47);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(18,26)(19,29)(20,32)(27,34)(30,36)(33,35)(37,47)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,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, 106612);