#9815094

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9722290);