#348667

order	512
nilpotency class	3
isoclinism	isoclinic class of #348654
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	#19004	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

749 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255863);