#8932746

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	#35420	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839942);