#8461590

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	#38123	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=h,e^2=ghi,f^2=i,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, 8368786);