#9863106

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

3733 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=gi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9770302);