#8898941

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

4006 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:

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

Polycyclic presentation:

2^9|a^2=hi,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, 8806137);