#347017

order	512
nilpotency class	3
isoclinism	isoclinic class of #347002
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62878	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254213);