#8228532

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	#56302	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=ghi,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,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8135728);