#8425436

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	#39277	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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)(3,10,8,4)(6,9,15,13)(11,14,16,12)(17,18,22,23)(19,25,20,24)(21,26,30,29)(27,32,28,31)(33,34,39,45)(35,40,37,42)(36,44,47,41)(38,46,48,43);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,14)(15,16)(18,23)(24,25)(26,29)(31,32)(33,35)(34,40)(36,46)(37,39)(38,41)(42,45)(43,47)(44,48);(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,4,7,10)(2,8,5,3)(6,14,15,12)(9,11,13,16)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,37,39,35)(34,42,45,40)(36,46,47,43)(38,44,48,41);(1,5,7,2)(3,10,8,4)(6,13,15,9)(11,14,16,12)(17,20,22,19)(18,25,23,24)(21,28,30,27)(26,32,29,31)(33,38)(34,43)(35,41)(36,42)(37,44)(39,48)(40,47)(45,46);(1,6)(2,9)(3,11)(4,12)(5,13)(7,15)(8,16)(10,14)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,36)(34,44)(35,46)(37,43)(38,42)(39,47)(40,48)(41,45)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=gh,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, 8332632);