#8097207

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,10)(4,9)(5,8)(6,11)(7,12)(13,15)(14,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,3,7,5)(2,8,12,10)(4,13,14,6)(9,11,16,15)(17,19,24,22)(18,25,21,20)(23,27,31,30)(26,32,29,28)(35,41)(39,42)(43,47)(46,48);(1,4,7,14)(2,9,12,16)(3,13,5,6)(8,11,10,15)(17,20)(18,22)(19,21)(23,28)(24,25)(26,30)(27,29)(31,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,7,3)(2,10,12,8)(4,6,14,13)(9,15,16,11)(17,21,24,18)(19,25,22,20)(23,29,31,26)(27,32,30,28);(1,6)(2,11)(3,4)(5,14)(7,13)(8,16)(9,10)(12,15)(17,22,24,19)(18,25,21,20)(23,27,31,30)(26,28,29,32)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,2)(3,8)(4,9)(5,10)(6,15)(7,12)(11,13)(14,16)(17,23,24,31)(18,26,21,29)(19,27,22,30)(20,28,25,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=g,e^2=i,f^2=i,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, 8004403);