#8097094

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

3879 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004290);