#8092717

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	#52391	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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:

(1,2,6,10)(3,9,5,7)(4,8,13,16)(11,15,12,14)(17,18,22,24)(19,27,30,23)(20,26,21,25)(28,31,29,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,3,6,5)(2,7,10,9)(4,11,13,12)(8,14,16,15)(17,19,22,30)(18,23,24,27)(20,29,21,28)(25,32,26,31)(33,35,37,42)(34,38,40,46)(36,43,44,41)(39,47,48,45);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(18,24)(23,27)(25,26)(31,32)(33,36)(34,39)(35,41)(37,44)(38,45)(40,48)(42,43)(46,47);(1,3,6,5)(2,7,10,9)(4,12,13,11)(8,15,16,14)(17,20,22,21)(18,25,24,26)(19,28,30,29)(23,31,27,32)(36,44)(39,48)(41,43)(45,47);(1,5,6,3)(2,9,10,7)(4,12,13,11)(8,15,16,14)(17,20,22,21)(18,26,24,25)(19,29,30,28)(23,31,27,32)(34,40)(38,46)(39,48)(45,47);(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=gi,d^2=g,e^2=g,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, 7999913);