#10106233

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	#51758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(18,19)(23,25)(24,26)(31,32)(33,34,38,41)(35,39,45,48)(36,37,46,40)(42,47,43,44);(2,7)(4,15)(8,11)(10,13)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32)(33,35,38,45)(34,39,41,48)(36,43,46,42)(37,44,40,47);(2,7)(3,12)(9,14)(10,13)(33,36,38,46)(34,40,41,37)(35,42,45,43)(39,44,48,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,36)(34,37)(35,43)(38,46)(39,44)(40,41)(42,45)(47,48);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,35)(34,39)(36,42)(37,47)(38,45)(40,44)(41,48)(43,46);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,37)(34,36)(35,44)(38,40)(39,43)(41,46)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10013429);