#10444979

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=i,e^2=gi,f^2=h,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, 10352175);