#8755237

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3845 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)(3,5)(4,12)(6,7)(8,11)(9,13)(10,15)(14,16)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(2,7)(4,13)(5,8)(10,16)(18,23)(20,25)(24,29)(28,31)(37,45)(40,44)(42,48)(46,47);(1,3,6,11)(2,5,7,8)(4,10,13,16)(9,14,12,15)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,14)(13,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4,6,13)(2,9,7,12)(3,10,11,16)(5,14,8,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47);(1,5,6,8)(2,3,7,11)(4,14,13,15)(9,10,12,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(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=h,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, 8662433);