#8430577

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

4069 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,f^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, 8337773);