#8932107

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	#45830	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

(1,2,7,11)(3,8,10,5)(4,13,6,9)(12,16,14,15)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(33,34,39,36)(35,38,43,40)(37,41,45,44)(42,46,48,47);(1,3,7,10)(2,8,11,5)(4,14,6,12)(9,16,13,15)(18,23)(20,25)(24,29)(28,31)(37,45)(41,44)(42,48)(46,47);(1,4,7,6)(2,9,11,13)(3,12,10,14)(5,16,8,15)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,30)(31,32)(33,35,39,43)(34,40,36,38)(37,42,45,48)(41,47,44,46);(1,5,7,8)(2,10,11,3)(4,15,6,16)(9,12,13,14)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,36,39,34)(35,38,43,40)(37,44,45,41)(42,46,48,47);(1,6,7,4)(2,9,11,13)(3,12,10,14)(5,15,8,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48);(1,2)(3,8)(4,13)(5,10)(6,9)(7,11)(12,16)(14,15)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(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=gi,c^2=gh,d^2=gh,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, 8839303);