#10481060

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	#56353	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388256);