#8811228

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	#50115	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=gh,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8718424);