#347179

order	512
nilpotency class	3
isoclinism	isoclinic class of #347154
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	#62886	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2094 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:

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

Polycyclic presentation:

2^9|a^2=i,b^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,e^c=ei,f^c=fi,g^c=gh,f^d=fh

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, 254375);