#8425413

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332609);