#8696003

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	#57144	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603199);