#8131183

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	#48341	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4209 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=hi,e^2=hi,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, 8038379);