#8324631

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	#40036	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=gh,f^2=hi,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, 8231827);