#8096888

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	#52404	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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)(3,10,8,4)(6,9,14,13)(11,16,15,12)(17,18,23,22)(19,26,24,20)(21,29,31,25)(27,28,30,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(17,19)(18,20)(21,30)(22,26)(23,24)(25,32)(27,31)(28,29)(35,42)(39,46)(43,47)(44,48);(17,20)(18,24)(19,22)(21,28)(23,26)(25,30)(27,29)(31,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,4,7,10)(2,8,5,3)(6,12,14,16)(9,15,13,11)(17,21,23,31)(18,25,22,29)(19,27,24,30)(20,28,26,32);(1,5,7,2)(3,10,8,4)(6,9,14,13)(11,12,15,16)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,30)(31,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22,23,18)(19,26,24,20)(21,29,31,25)(27,32,30,28)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=ghi,d^2=g,e^2=gh,f^2=h,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, 8004084);