#8105158

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#53673	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8012354);