#213988

order	512
nilpotency class	3
isoclinism	isoclinic class of #213876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

903 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121184);