#347009

order	512
nilpotency class	3
isoclinism	isoclinic class of #347002
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	#18304	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1961 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:

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

Polycyclic presentation:

2^9|b^2=h,c^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,g^b=gi,d^c=dh,e^c=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254205);