#8700281

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3873 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=g,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8607477);