#8430498

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4214 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8337694);