#8093105

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	#52459	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)(3,8)(4,13)(5,9)(6,7)(10,14)(11,16)(12,15)(17,18)(19,26)(20,30)(21,24)(22,25)(23,29)(27,31)(28,32)(33,34)(35,39)(36,45)(37,40)(38,41)(42,48)(43,46)(44,47);(2,7)(8,14)(9,13)(15,16)(17,19,22,29)(18,23,25,26)(20,27,21,28)(24,31,30,32)(33,35,38,44)(34,39,41,47)(36,43,37,42)(40,48,45,46);(1,3)(2,8)(4,11)(5,12)(6,10)(7,14)(9,15)(13,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,26,32)(33,36,38,37)(34,40,41,45)(35,42,44,43)(39,46,47,48);(2,7)(3,10)(9,13)(11,12)(17,20,22,21)(18,24,25,30)(19,28,29,27)(23,32,26,31)(33,37,38,36)(34,40,41,45)(35,43,44,42)(39,46,47,48);(1,4,6,5)(2,9,7,13)(3,11,10,12)(8,15,14,16)(17,21,22,20)(18,24,25,30)(19,28,29,27)(23,31,26,32);(1,5,6,4)(2,9,7,13)(3,12,10,11)(8,15,14,16)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=g,f^2=ghi,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, 8000301);