#8271889

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	#44147	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(3,8,14,9)(4,15,5,6)(11,13,12,16)(33,34,40,44)(35,45,47,41)(36,38,37,42)(39,43,46,48);(1,3,7,14)(2,8,10,9)(4,12,5,11)(6,13,15,16)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(33,35)(34,41)(36,43)(37,48)(38,39)(40,47)(42,46)(44,45);(1,4,7,5)(2,6,10,15)(3,11,14,12)(8,16,9,13)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,36,40,37)(34,42,44,38)(35,43,47,48)(39,41,46,45);(1,5,7,4)(2,6,10,15)(3,12,14,11)(8,16,9,13)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,37,40,36)(34,38,44,42)(35,43,47,48)(39,41,46,45);(1,3)(2,9)(4,11)(5,12)(6,13)(7,14)(8,10)(15,16)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32)(33,38)(34,36)(35,46)(37,44)(39,47)(40,42)(41,48)(43,45);(1,6)(2,4)(3,13)(5,10)(7,15)(8,12)(9,11)(14,16)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)(33,39,40,46)(34,43,44,48)(35,38,47,42)(36,45,37,41)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8179085);