#8931293

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	#53506	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=gi,f^2=gh,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, 8838489);