#8322200

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	#42471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8229396);