#346981

order	512
nilpotency class	3
isoclinism	isoclinic class of #346950
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	#62850	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2079 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,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,e^c=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254177);