#8434123

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	#52471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4247 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)(4,9)(5,14)(6,8)(7,11)(12,16)(13,15)(20,28)(24,31)(27,32)(29,30)(33,34,40,38)(35,44,46,41)(36,47,42,37)(39,43,45,48);(1,3,7,6)(2,8,11,10)(4,12,5,13)(9,15,14,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,35)(34,41)(36,45)(37,48)(38,44)(39,42)(40,46)(43,47);(1,4,7,5)(2,9,11,14)(3,12,6,13)(8,15,10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,37)(35,45)(38,47)(39,46)(40,42)(41,48)(43,44);(1,5,7,4)(2,9,11,14)(3,13,6,12)(8,15,10,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,37,40,47)(34,42,38,36)(35,43,46,48)(39,41,45,44);(1,6,7,3)(2,10,11,8)(4,13,5,12)(9,16,14,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,32)(26,31)(33,38,40,34)(35,44,46,41)(36,47,42,37)(39,48,45,43);(1,2)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,39)(34,43)(35,42)(36,46)(37,44)(38,48)(40,45)(41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=g,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, 8341319);