#8127597

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	#51126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=g,f^2=h,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, 8034793);