#9039159

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

3769 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=gi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,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, 8946355);