#8926703

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	#57170	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833899);