#8648931

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3869 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8556127);