#8093046

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
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	#53062	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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:

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

Polycyclic presentation:

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

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, 8000242);