#8656223

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#45524	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8563419);