#8436615

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=h,e^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, 8343811);