#8097317

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

2^9|a^2=gi,d^2=gh,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, 8004513);