#10352528

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	#54021	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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:

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

Polycyclic presentation:

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

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, 10259724);