#8096508

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

3875 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=gi,f^2=gi,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, 8003704);