#10320359

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	#43429	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227555);