#10506285

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	#55689	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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)(4,7)(5,8)(6,12)(9,14)(11,15)(13,16)(17,18,23,21)(19,24,30,29)(20,26,31,25)(22,32,28,27)(33,34)(35,36)(37,41)(38,42)(39,40)(43,44)(45,48)(46,47);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,19,23,30)(18,24,21,29)(20,28,31,22)(25,32,26,27)(34,40)(36,41)(42,47)(44,48);(1,2,7,4)(3,10,5,8)(6,9,14,12)(11,16,13,15)(17,20)(18,25)(19,28)(21,26)(22,30)(23,31)(24,32)(27,29)(33,35,39,37)(34,36,40,41)(38,43,46,45)(42,44,47,48);(1,4,7,2)(3,10,5,8)(6,12,14,9)(11,16,13,15)(17,20)(18,26)(19,28)(21,25)(22,30)(23,31)(24,27)(29,32)(33,36,39,41)(34,35,40,37)(38,44,46,48)(42,43,47,45);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,21,23,18)(19,29,30,24)(20,26,31,25)(22,32,28,27)(33,37,39,35)(34,41,40,36)(38,43,46,45)(42,44,47,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22,23,28)(18,27,21,32)(19,20,30,31)(24,25,29,26)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=ghi,f^2=gh,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, 10413481);