#246950

order	512
nilpotency class	3
isoclinism	isoclinic class of #246927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1066 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)(17,18)(19,28)(20,24)(21,25)(22,27)(23,30)(26,35)(29,44)(31,42)(32,39)(33,41)(34,37)(36,47)(38,45)(40,48)(43,46);(1,2,6,4)(3,7,11,10)(5,12,13,8)(9,16,14,15)(17,19)(18,23)(20,29)(21,34)(22,31)(24,38)(25,28)(26,42)(27,37)(30,35)(32,47)(33,46)(36,41)(39,44)(40,43)(45,48);(1,3)(2,7)(4,10)(5,14)(6,11)(8,16)(9,13)(12,15)(17,20,22,33)(18,24,27,41)(19,29,31,46)(21,32,35,48)(23,36,37,38)(25,39,26,40)(28,43,42,44)(30,45,34,47);(1,4,6,2)(3,10,11,7)(5,8,13,12)(9,15,14,16)(18,25)(21,35)(23,28)(24,39)(26,27)(30,34)(32,48)(36,43)(37,42)(38,44)(40,41)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154146);