#8932127

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	#56665	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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=gh,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, 8839323);