#8534776

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	#43080	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4244 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gi,d^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8441972);