#9863328

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	#57350	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 9770524);