#8497822

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3892 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=h,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, 8405018);