#8455708

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,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, 8362904);