#280942

order	512
nilpotency class	3
isoclinism	isoclinic class of #280915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

475 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,24,30)(19,27,33,41)(20,34,38,44)(21,39,32,25)(22,29,23,28)(26,37,43,36)(31,45,47,40)(35,48,46,42);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,13)(15,16)(17,19,23,21,24,33,22,32)(18,25,29,41,30,39,28,27)(20,35,36,31,38,46,37,47)(26,42,34,45,43,48,44,40);(3,10)(5,13)(6,14)(8,16)(9,15)(11,12)(17,20)(18,26)(19,31)(21,35)(22,36)(23,37)(24,38)(25,40)(27,42)(28,34)(29,44)(30,43)(32,46)(33,47)(39,45)(41,48);(2,7)(6,9)(8,16)(14,15)(17,21,23,33,24,32,22,19)(18,27,29,25,30,41,28,39)(20,35,37,47,38,46,36,31)(26,42,44,40,43,48,34,45)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=ehi,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=ghi

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, 188138);