#8105496

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	#53658	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4068 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)(3,10,12,8)(5,14,6,9)(11,15,13,16)(17,18)(19,22)(20,25)(21,26)(23,28)(24,30)(27,31)(29,32)(33,34,37,40)(35,38,42,47)(36,43,45,39)(41,48,44,46);(1,3,7,12)(2,8,4,10)(5,11,6,13)(9,15,14,16)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,4,7,2)(3,10,12,8)(5,9,6,14)(11,15,13,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36)(34,39)(35,41)(37,45)(38,46)(40,43)(42,44)(47,48);(1,5,7,6)(2,9,4,14)(3,11,12,13)(8,15,10,16)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(35,42)(36,45)(38,47)(39,43);(3,12)(8,10)(11,13)(15,16)(18,19)(23,25)(24,26)(31,32)(34,40)(35,42)(39,43)(41,44);(1,6,7,5)(2,9,4,14)(3,13,12,11)(8,15,10,16)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,f^2=ghi,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, 8012692);