#10446443

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4140 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=gi,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, 10353639);