#8402273

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

3990 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8309469);