#8926536

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	#36923	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833732);