#10451579

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	#41324	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3966 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)(3,5)(4,11)(6,9)(7,8)(10,13)(12,16)(14,15)(17,18,24,26)(19,22,23,32)(20,28,21,25)(27,31,29,30)(33,34,39,41)(35,42,37,40)(36,38,45,48)(43,47,46,44);(2,8)(5,11)(9,15)(13,16)(17,19,24,23)(18,22,26,32)(20,29,21,27)(25,31,28,30);(1,3,7,4)(2,5,8,11)(6,10,14,12)(9,13,15,16)(17,20,24,21)(18,25,26,28)(19,27,23,29)(22,30,32,31)(33,35,39,37)(34,40,41,42)(36,43,45,46)(38,44,48,47);(1,4,7,3)(2,5,8,11)(6,10,14,12)(9,16,15,13)(17,21,24,20)(18,25,26,28)(19,27,23,29)(22,31,32,30)(33,36)(34,38)(35,43)(37,46)(39,45)(40,44)(41,48)(42,47);(1,5)(2,4)(3,8)(6,13)(7,11)(9,12)(10,15)(14,16)(17,22)(18,19)(20,30)(21,31)(23,26)(24,32)(25,27)(28,29)(33,37,39,35)(34,40,41,42)(36,43,45,46)(38,47,48,44);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,23,24,19)(18,22,26,32)(20,29,21,27)(25,30,28,31)(33,38)(34,36)(35,44)(37,47)(39,48)(40,43)(41,45)(42,46)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10358775);