#8504746

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3822 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=i,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411942);