#8701725

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=hi,e^2=ghi,f^2=gi,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, 8608921);