#8630918

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	#49690	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4103 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)(3,8,4,11)(6,9,14,13)(10,15,12,16)(17,18,24,29)(19,22,31,25)(20,27,21,26)(23,28,30,32)(33,34,38,41)(35,37,40,47)(36,43,46,39)(42,48,44,45);(2,5)(8,11)(9,13)(15,16)(17,19)(18,25)(20,30)(21,23)(22,29)(24,31)(26,32)(27,28)(33,35,38,40)(34,37,41,47)(36,44,46,42)(39,48,43,45);(1,3,7,4)(2,8,5,11)(6,10,14,12)(9,15,13,16)(17,20,24,21)(18,26,29,27)(19,30,31,23)(22,28,25,32)(33,36,38,46)(34,39,41,43)(35,42,40,44)(37,45,47,48);(1,4,7,3)(2,8,5,11)(6,10,14,12)(9,16,13,15)(17,21,24,20)(18,27,29,26)(19,30,31,23)(22,28,25,32)(33,35)(34,37)(36,42)(38,40)(39,45)(41,47)(43,48)(44,46);(1,5,7,2)(3,8,4,11)(6,13,14,9)(10,15,12,16)(17,22)(18,19)(20,32)(21,28)(23,26)(24,25)(27,30)(29,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,43)(47,48);(1,6,7,14)(2,9,5,13)(3,10,4,12)(8,15,11,16)(17,23,24,30)(18,28,29,32)(19,21,31,20)(22,26,25,27)(33,37)(34,40)(35,41)(36,45)(38,47)(39,44)(42,43)(46,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=g,e^2=ghi,f^2=g,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, 8538114);