#8425450

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	#37484	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4146 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)(3,9)(4,10)(5,8)(6,7)(11,15)(12,16)(13,14)(17,18,22,23)(19,26,21,24)(20,28,30,25)(27,31,29,32)(33,34,39,45)(35,40,37,42)(36,44,47,41)(38,46,48,43);(2,7)(3,10)(8,14)(11,16)(18,23)(24,26)(25,28)(31,32)(33,35)(34,40)(36,46)(37,39)(38,41)(42,45)(43,47)(44,48);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,19,22,21)(18,24,23,26)(20,27,30,29)(25,31,28,32)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32)(33,37,39,35)(34,42,45,40)(36,46,47,43)(38,44,48,41);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,15)(14,16)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)(33,38)(34,43)(35,41)(36,42)(37,44)(39,48)(40,47)(45,46);(1,5,6,13)(2,8,7,14)(3,11,10,16)(4,12,9,15)(17,21,22,19)(18,24,23,26)(20,29,30,27)(25,31,28,32)(33,36)(34,44)(35,46)(37,43)(38,42)(39,47)(40,48)(41,45)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=gh,f^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, 8332646);