#8455730

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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)(3,10)(4,13)(5,14)(6,8)(7,9)(11,12)(15,16)(17,18,24,22)(19,29,30,25)(20,27,21,26)(23,28,32,31)(33,34,38,37)(35,41,42,39)(36,40,44,45)(43,47,48,46);(1,3,7,6)(2,8,9,10)(4,11,14,16)(5,12,13,15)(17,19)(18,25)(20,28)(21,31)(22,29)(23,27)(24,30)(26,32)(35,42)(36,44)(39,41)(40,45);(17,20,24,21)(18,26,22,27)(19,28,30,31)(23,25,32,29)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,4)(2,5)(3,11)(6,16)(7,14)(8,12)(9,13)(10,15)(17,21,24,20)(18,27,22,26)(19,28,30,31)(23,25,32,29)(33,36,38,44)(34,40,37,45)(35,43,42,48)(39,46,41,47);(1,5)(2,4)(3,12)(6,15)(7,13)(8,11)(9,14)(10,16)(17,22,24,18)(19,29,30,25)(20,27,21,26)(23,31,32,28);(1,6,7,3)(2,8,9,10)(4,11,14,16)(5,15,13,12)(17,23)(18,28)(19,27)(20,25)(21,29)(22,31)(24,32)(26,30)(33,37,38,34)(35,41,42,39)(36,40,44,45)(43,46,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=h,e^2=hi,f^2=gh,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, 8362926);