#280939

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	#28174	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

494 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:

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

Polycyclic presentation:

2^9|b^2=f,d^2=ei,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, 188135);