#8745737

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	#52407	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=ghi,d^2=gi,f^2=gh,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, 8652933);