#9863401

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	#52380	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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)(3,11)(4,7)(5,9)(6,15)(8,13)(10,16)(12,14)(17,18,23,28)(19,24,30,21)(20,26,32,25)(22,31,29,27)(34,40)(36,45)(41,46)(43,44);(1,3,7,13)(2,8,4,11)(5,6,12,16)(9,10,14,15)(17,19,23,30)(18,24,28,21)(20,29,32,22)(25,31,26,27)(33,34)(35,41)(36,42)(37,46)(38,43)(39,40)(44,48)(45,47);(2,4)(8,11)(9,14)(10,15)(17,20,23,32)(18,25,28,26)(19,22,30,29)(21,31,24,27)(33,35,39,37)(34,41,40,46)(36,44,45,43)(38,42,48,47);(1,4,7,2)(3,11,13,8)(5,14,12,9)(6,15,16,10)(17,20)(18,26)(19,29)(21,31)(22,30)(23,32)(24,27)(25,28)(33,36)(34,42)(35,44)(37,43)(38,46)(39,45)(40,47)(41,48);(1,5)(2,9)(3,6)(4,14)(7,12)(8,10)(11,15)(13,16)(17,21)(18,19)(20,31)(22,25)(23,24)(26,29)(27,32)(28,30)(33,37,39,35)(34,41,40,46)(36,43,45,44)(38,42,48,47);(1,6)(2,10)(3,12)(4,15)(5,13)(7,16)(8,14)(9,11)(17,22,23,29)(18,27,28,31)(19,20,30,32)(21,26,24,25)(33,38,39,48)(34,43,40,44)(35,42,37,47)(36,46,45,41)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=hi,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, 9770597);