#8755238

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	#54156	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662434);