#8095388

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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)(3,10,4,8)(6,13,14,9)(11,15,12,16)(21,29)(24,28)(26,32)(30,31)(33,34,39,44)(35,37,42,40)(36,43,38,41)(45,48,46,47);(1,2,7,5)(3,8,4,10)(6,9,14,13)(11,15,12,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,35)(34,40)(36,45)(37,44)(38,46)(39,42)(41,48)(43,47);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35)(34,37)(36,45)(38,46)(39,42)(40,44)(41,47)(43,48);(1,4,7,3)(2,8,5,10)(6,12,14,11)(9,15,13,16)(33,36,39,38)(34,41,44,43)(35,45,42,46)(37,47,40,48);(1,5,7,2)(3,8,4,10)(6,13,14,9)(11,15,12,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,37)(34,42)(35,44)(36,47)(38,48)(39,40)(41,46)(43,45);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,36)(34,43,44,41)(35,46,42,45)(37,48,40,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=gh,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002584);