#348052

order	512
nilpotency class	3
isoclinism	isoclinic class of #348038
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

840 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18,23,27)(19,28,26,22)(20,31,35,24)(21,25,37,40)(29,34,32,41)(30,42,39,36)(33,38,47,44)(43,48,45,46);(2,6)(7,15)(8,13)(9,16)(17,19,22,18,23,26,28,27)(20,32,34,31,35,29,41,24)(21,30,36,25,37,39,42,40)(33,45,46,44,47,43,48,38);(1,3)(2,7)(4,10)(5,14)(6,15)(8,16)(9,13)(11,12)(17,20)(18,24)(19,29)(21,33)(22,34)(23,35)(25,38)(26,32)(27,31)(28,41)(30,43)(36,46)(37,47)(39,45)(40,44)(42,48);(5,12)(8,13)(9,16)(11,14)(17,21)(18,25)(19,30)(20,33)(22,36)(23,37)(24,38)(26,39)(27,40)(28,42)(29,43)(31,44)(32,45)(34,46)(35,47)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255248);