#8295508

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	#58484	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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)(3,10,5,8)(4,12,14,9)(11,15,13,16)(33,34,38,39)(35,36,41,44)(37,40,45,46)(42,43,47,48);(1,3,7,5)(2,8,6,10)(4,13,14,11)(9,16,12,15)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32)(34,39)(35,41)(40,46)(42,47);(2,6)(8,10)(9,12)(15,16)(18,23)(21,30)(24,31)(28,32)(33,35)(34,36)(37,42)(38,41)(39,44)(40,43)(45,47)(46,48);(1,4,7,14)(2,9,6,12)(3,11,5,13)(8,15,10,16)(17,19,22,25)(18,21,23,30)(20,27,29,26)(24,32,31,28)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,48)(46,47);(1,5,7,3)(2,10,6,8)(4,11,14,13)(9,15,12,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,31)(25,27)(30,32);(1,6,7,2)(3,10,5,8)(4,12,14,9)(11,16,13,15)(17,21)(18,25)(19,23)(20,28)(22,30)(24,27)(26,31)(29,32)(33,37,38,45)(34,40,39,46)(35,42,41,47)(36,43,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gi,e^2=h,f^2=g,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, 8202704);