#9937778

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	#56695	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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)(3,7)(4,10)(5,13)(6,8)(9,12)(11,16)(14,15)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(33,34,38,40)(35,36,41,43)(37,39,46,48)(42,44,45,47);(1,3,6,12)(2,7,8,9)(4,13,15,11)(5,14,16,10)(18,20)(24,26)(25,28)(31,32)(35,41)(36,43)(42,45)(44,47);(2,8)(3,12)(10,14)(11,13)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,46)(47,48);(1,3)(2,9)(4,13)(5,14)(6,12)(7,8)(10,16)(11,15)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32)(33,36)(34,35)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47);(1,4)(2,10)(3,13)(5,7)(6,15)(8,14)(9,16)(11,12)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,35)(34,36)(37,45)(38,41)(39,47)(40,43)(42,46)(44,48);(1,5,6,16)(2,11,8,13)(3,14,12,10)(4,9,15,7)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,37)(34,39)(35,42)(36,44)(38,46)(40,48)(41,45)(43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9844974);