#8127536

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	#56376	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034732);