#10320372

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	#44438	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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18,23,25)(19,28,29,24)(20,30,27,22)(21,26,31,32)(33,34)(35,44)(36,40)(37,48)(38,43)(39,46)(41,47)(42,45);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19)(18,24)(20,21)(22,26)(23,29)(25,28)(27,31)(30,32)(33,35)(34,39)(36,37)(38,46)(40,42)(41,48)(43,44)(45,47);(2,7)(6,13)(8,11)(14,15)(18,25)(19,29)(20,27)(26,32)(33,36,38,47)(34,40,43,41)(35,37,46,45)(39,42,44,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,23,27)(18,22,25,30)(19,21,29,31)(24,26,28,32)(33,36)(34,41)(35,45)(37,46)(38,47)(39,42)(40,43)(44,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,26)(19,27)(20,29)(22,28)(23,31)(24,30)(25,32)(33,34)(35,44)(36,40)(37,48)(38,43)(39,46)(41,47)(42,45);(1,4,5,12)(2,8,7,11)(3,9,10,16)(6,14,13,15)(17,22)(18,27)(19,26)(20,25)(21,28)(23,30)(24,31)(29,32)(33,37,38,45)(34,42,43,48)(35,36,46,47)(39,40,44,41)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gi,d^2=i,f^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, 10227568);