#10451596

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	#41158	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=gh,d^2=hi,e^2=hi,f^2=i,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, 10358792);