#8932096

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	#53838	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3906 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=h,e^2=i,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, 8839292);