#9863088

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	#55039	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4023 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,21,23)(19,25,27,22)(20,24,29,31)(26,32,28,30)(33,34,40,44)(35,41,37,38)(36,43,39,42)(45,47,46,48);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32)(33,35,40,37)(34,41,44,38)(36,46,39,45)(42,47,43,48);(2,4)(3,10)(9,12)(11,16)(18,23)(22,25)(24,31)(30,32)(33,36,40,39)(34,42,44,43)(35,45,37,46)(38,47,41,48);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)(33,37,40,35)(34,38,44,41)(36,45,39,46)(42,48,43,47);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,20,21,29)(18,24,23,31)(19,26,27,28)(22,30,25,32)(33,38)(34,35)(36,47)(37,44)(39,48)(40,41)(42,45)(43,46);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,19)(18,22)(20,28)(21,27)(23,25)(24,32)(26,29)(30,31)(33,39,40,36)(34,43,44,42)(35,45,37,46)(38,47,41,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=i,e^2=hi,f^2=gh,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, 9770284);