#8932191

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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)(3,8)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(34,39)(37,40)(41,46)(45,47);(1,3,7,4)(2,8,9,5)(6,11,15,12)(10,16,14,13)(18,23)(20,25)(24,29)(28,31)(36,44)(41,46)(43,48)(45,47);(2,9)(3,4)(6,15)(13,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,4,7,3)(2,8,9,5)(6,12,15,11)(10,16,14,13)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,5,7,8)(2,4,9,3)(6,13,15,16)(10,12,14,11)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,6,7,15)(2,10,9,14)(3,11,4,12)(5,13,8,16)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=gh,e^2=gi,f^2=h,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, 8839387);