#8505078

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	#40164	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4204 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)(3,8)(4,9)(5,10)(6,11)(7,12)(13,16)(14,15)(17,18,23,28)(19,25,20,24)(21,22,27,26)(29,30,31,32)(33,34)(35,43)(36,37)(38,40)(39,42)(41,47)(44,46)(45,48);(3,12)(4,14)(7,8)(9,15)(17,19,23,20)(18,24,28,25)(21,31,27,29)(22,30,26,32)(33,35,39,38)(34,40,42,43)(36,44,47,45)(37,48,41,46);(1,3)(2,7)(4,13)(5,9)(6,12)(8,11)(10,14)(15,16)(17,19,23,20)(18,25,28,24)(21,29,27,31)(22,30,26,32)(33,36)(34,41)(35,44)(37,42)(38,45)(39,47)(40,46)(43,48);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,16)(10,14)(17,20,23,19)(18,25,28,24)(21,31,27,29)(22,30,26,32)(33,36)(34,41)(35,45)(37,42)(38,44)(39,47)(40,48)(43,46);(1,4)(2,9)(3,13)(5,7)(6,14)(8,16)(10,12)(11,15)(17,21,23,27)(18,26,28,22)(19,29,20,31)(24,30,25,32)(33,37,39,41)(34,36,42,47)(35,46,38,48)(40,45,43,44);(1,5,6,16)(2,10,11,13)(3,9,12,15)(4,7,14,8)(17,22,23,26)(18,27,28,21)(19,30,20,32)(24,29,25,31)(33,38,39,35)(34,40,42,43)(36,45,47,44)(37,48,41,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=ghi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412274);