#8755001

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

3771 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:

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

Polycyclic presentation:

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

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, 8662197);