#8455707

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=h,e^2=hi,f^2=h,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, 8362903);