#8496828

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	#37946	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=h,f^2=gi,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, 8404024);