#347160

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1996 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:

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

Polycyclic presentation:

2^9|b^2=i,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, 254356);