#8119201

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	#43192	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3996 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=gi,e^2=gh,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^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026397);