#8425432

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	#38132	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4185 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8332628);