#8745987

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653183);