#8131195

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	#57981	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4207 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8038391);