#8093096

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	#52499	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4072 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)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18,22,25)(19,26,28,23)(20,30,21,24)(27,31,29,32)(33,34,37,40)(35,38,42,47)(36,43,45,39)(41,48,44,46);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19,22,28)(18,23,25,26)(20,27,21,29)(24,31,30,32)(33,35)(34,38)(36,44)(37,42)(39,48)(40,47)(41,45)(43,46);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48);(2,9)(4,12)(7,10)(11,13)(18,25)(19,28)(24,30)(27,29)(35,42)(38,47)(41,44)(46,48);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,21,22,20)(18,24,25,30)(19,29,28,27)(23,31,26,32);(1,5,6,3)(2,7,9,10)(4,13,12,11)(8,14,15,16)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^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, 8000292);