#8096216

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(19,22)(20,24)(21,25)(23,28)(26,30)(27,31)(29,32)(33,34,39,38)(35,42,40,36)(37,45,47,41)(43,44,46,48);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30)(33,35)(34,36)(37,46)(38,42)(39,40)(41,48)(43,47)(44,45);(17,20)(18,23)(19,26)(21,28)(22,29)(24,25)(27,32)(30,31)(33,36)(34,40)(35,38)(37,44)(39,42)(41,46)(43,45)(47,48);(1,4,7,3)(2,9,5,8)(6,12,14,11)(10,16,13,15)(17,19)(18,22)(20,26)(21,27)(23,29)(24,30)(25,31)(28,32)(33,37,39,47)(34,41,38,45)(35,43,40,46)(36,44,42,48);(1,5,7,2)(3,9,4,8)(6,10,14,13)(11,15,12,16)(17,20)(18,24)(19,26)(21,28)(22,30)(23,25)(27,32)(29,31)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,46)(47,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,18)(19,22)(20,23)(21,25)(24,28)(26,29)(27,31)(30,32)(33,38,39,34)(35,42,40,36)(37,45,47,41)(43,48,46,44)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=gi,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, 8003412);