#8472384

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	#55706	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3951 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8379580);