#8898936

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4002 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)(3,12)(4,15)(5,6)(7,9)(8,14)(10,13)(11,16)(17,18,23,25)(19,24,29,30)(20,26,28,22)(21,27,31,32)(33,34,38,41)(35,39,45,42)(36,37,40,46)(43,44,47,48);(1,3,7,14)(2,8,9,12)(4,11,13,5)(6,10,16,15)(17,19)(18,24)(20,27)(21,22)(23,29)(25,30)(26,31)(28,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(2,9)(6,16)(8,12)(10,15)(18,25)(19,29)(22,26)(27,32)(33,34,38,41)(35,42,45,39)(36,37,40,46)(43,48,47,44);(1,4)(2,10)(3,11)(5,14)(6,12)(7,13)(8,16)(9,15)(17,20,23,28)(18,26,25,22)(19,27,29,32)(21,24,31,30)(33,36)(34,37)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,5,7,11)(2,6,9,16)(3,13,14,4)(8,15,12,10)(17,21)(18,27)(19,26)(20,30)(22,29)(23,31)(24,28)(25,32)(33,37)(34,40)(35,44)(36,41)(38,46)(39,47)(42,43)(45,48);(1,6)(2,11)(3,10)(4,12)(5,9)(7,16)(8,13)(14,15)(17,22,23,26)(18,28,25,20)(19,21,29,31)(24,32,30,27)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8806132);