#10195864

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

3844 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)(3,12)(4,10)(5,6)(7,9)(8,11)(13,15)(14,16)(17,18,21,24)(19,25,27,22)(20,23,30,28)(26,32,29,31)(33,34)(35,41)(36,46)(37,39)(38,42)(40,43)(44,47)(45,48);(1,3,8,7)(2,9,11,12)(4,13,5,14)(6,15,10,16)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,35,40,37)(34,41,43,39)(36,45,38,44)(42,47,46,48);(1,4,8,5)(2,10,11,6)(3,13,7,14)(9,16,12,15)(17,19,21,27)(18,22,24,25)(20,26,30,29)(23,31,28,32)(33,36,40,38)(34,42,43,46)(35,44,37,45)(39,47,41,48);(1,5,8,4)(2,10,11,6)(3,14,7,13)(9,16,12,15)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31)(33,37,40,35)(34,39,43,41)(36,44,38,45)(42,48,46,47);(1,6,8,10)(2,5,11,4)(3,15,7,16)(9,14,12,13)(17,20)(18,23)(19,26)(21,30)(22,31)(24,28)(25,32)(27,29)(33,38,40,36)(34,42,43,46)(35,45,37,44)(39,47,41,48);(1,7,8,3)(2,9,11,12)(4,14,5,13)(6,15,10,16)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,39)(34,35)(36,47)(37,43)(38,48)(40,41)(42,44)(45,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=gh,e^2=gh,f^2=ghi,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, 10103060);