#8932074

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3772 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)(3,7)(4,8)(5,9)(6,10)(11,15)(12,13)(14,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(1,3,6,8)(2,7,10,4)(5,13,14,11)(9,12,16,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,3)(2,7)(4,10)(5,11)(6,8)(9,15)(12,16)(13,14)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,5,6,14)(2,9,10,16)(3,11,8,13)(4,12,7,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839270);