#8436770

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	#55316	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(18,23)(19,28)(26,30)(27,32)(33,34)(35,43)(36,40)(37,41)(38,42)(39,44)(45,48)(46,47);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,18)(19,24)(20,26)(21,27)(22,23)(25,28)(29,30)(31,32)(35,44)(36,46)(39,43)(40,47);(17,19)(18,24)(20,27)(21,26)(22,28)(23,25)(29,32)(30,31)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,41,47,48);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,19)(18,25)(20,27)(21,30)(22,28)(23,24)(26,31)(29,32)(33,36)(34,40)(35,45)(37,44)(38,46)(39,41)(42,47)(43,48);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,20,22,29)(18,26,23,30)(19,27,28,32)(21,25,31,24)(33,37,38,45)(34,41,42,48)(35,36,44,46)(39,47,43,40);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,21,22,31)(18,27,23,32)(19,26,28,30)(20,25,29,24)(33,34)(35,39)(36,47)(37,41)(38,42)(40,46)(43,44)(45,48)

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343966);