#8471039

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	#42990	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4108 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378235);