#8184819

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	#42557	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092015);