#8749692

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

4095 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8656888);