#8105929

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(3,10)(4,14)(5,8)(6,9)(7,13)(11,15)(12,16)(17,18)(19,26)(20,30)(21,25)(22,24)(23,29)(27,31)(28,32)(33,34,37,40)(35,38,42,46)(36,39,44,48)(41,45,43,47);(1,3,6,13)(2,7,9,10)(4,11,5,12)(8,15,14,16)(17,19,22,29)(18,23,24,26)(20,28,21,27)(25,32,30,31)(33,35)(34,38)(36,43)(37,42)(39,47)(40,46)(41,44)(45,48);(18,24)(23,26)(25,30)(31,32)(33,36,37,44)(34,39,40,48)(35,41,42,43)(38,45,46,47);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,22,21)(18,25,24,30)(19,27,29,28)(23,31,26,32);(1,4,6,5)(2,8,9,14)(3,12,13,11)(7,16,10,15)(17,21,22,20)(18,25,24,30)(19,28,29,27)(23,31,26,32)(34,40)(35,42)(39,48)(41,43);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)

Polycyclic presentation:

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

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, 8013125);