#8505093

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	#40053	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)(3,9)(4,12)(5,14)(6,11)(7,8)(10,15)(13,16)(17,18,22,26)(19,23,29,24)(20,25,31,21)(27,28,30,32)(33,34)(35,43)(36,37)(38,40)(39,42)(41,47)(44,46)(45,48);(2,8)(5,15)(9,12)(11,13)(17,19,22,29)(18,23,26,24)(20,30,31,27)(21,28,25,32)(33,35,39,38)(34,40,42,43)(36,44,47,45)(37,48,41,46);(1,3,7,4)(2,9,8,12)(5,11,15,13)(6,10,16,14)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,32)(30,31)(33,36)(34,41)(35,44)(37,42)(38,45)(39,47)(40,46)(43,48);(1,4,7,3)(2,9,8,12)(5,13,15,11)(6,10,16,14)(17,19)(18,23)(20,27)(21,32)(22,29)(24,26)(25,28)(30,31)(33,36)(34,41)(35,45)(37,42)(38,44)(39,47)(40,48)(43,46);(1,5,7,15)(2,10,8,14)(3,11,4,13)(6,9,16,12)(17,20)(18,25)(19,27)(21,26)(22,31)(23,28)(24,32)(29,30)(33,37,39,41)(34,36,42,47)(35,46,38,48)(40,45,43,44);(1,6,7,16)(2,11,8,13)(3,10,4,14)(5,9,15,12)(17,21,22,25)(18,20,26,31)(19,28,29,32)(23,30,24,27)(33,38,39,35)(34,40,42,43)(36,45,47,44)(37,48,41,46)

Polycyclic presentation:

2^9|a^2=i,b^2=h,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, 8412289);