#8928027

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	#54815	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=h,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.

Relationships

maximal normal subgroup of	#72219279, #72219281, #72219283
maximal quotient of	#72219272

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8835223);