#8114375

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	#57411	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,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, 8021571);