#8751081

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	#42907	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 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)(3,12)(4,10)(5,13)(6,8)(7,9)(11,15)(14,16)(19,21)(20,24)(27,29)(28,30)(36,44)(40,47)(43,45)(46,48);(1,3,7,6)(2,8,9,12)(4,13,14,11)(5,10,15,16)(17,18,23,26)(19,20,21,24)(22,25,31,32)(27,28,29,30)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(2,9)(4,14)(8,12)(11,13)(17,19,23,21)(18,24,26,20)(22,27,31,29)(25,30,32,28)(33,35,38,37)(34,39,41,42)(36,45,44,43)(40,48,47,46);(1,4)(2,10)(3,11)(5,8)(6,13)(7,14)(9,16)(12,15)(17,20,23,24)(18,19,26,21)(22,30,31,28)(25,29,32,27)(33,36,38,44)(34,40,41,47)(35,43,37,45)(39,46,42,48);(1,5)(2,11)(3,10)(4,8)(6,16)(7,15)(9,13)(12,14)(17,21,23,19)(18,24,26,20)(22,27,31,29)(25,28,32,30)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,6,7,3)(2,8,9,12)(4,11,14,13)(5,10,15,16)(17,22,23,31)(18,25,26,32)(19,27,21,29)(20,28,24,30)(33,37,38,35)(34,39,41,42)(36,43,44,45)(40,48,47,46)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=i,e^2=gh,f^2=gi,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, 8658277);