#8932219

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

3784 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)(3,4)(5,13)(6,9)(7,11)(8,14)(10,15)(12,16)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34)(35,37)(36,44)(38,42)(39,41)(40,46)(43,47)(45,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,13)(14,16)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(33,34,38,42)(35,37,41,39)(36,40,46,44)(43,45,48,47);(1,2,6,9)(3,7,11,4)(5,13,14,8)(10,12,16,15)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,35,38,41)(34,39,42,37)(36,43,46,48)(40,47,44,45);(1,2)(3,7)(4,11)(5,8)(6,9)(10,15)(12,16)(13,14)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,48)(42,44);(1,4,6,7)(2,3,9,11)(5,12,14,15)(8,10,13,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,46)(45,48);(1,5,6,14)(2,8,9,13)(3,10,11,16)(4,12,7,15)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,48,44,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,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, 8839415);