#8092765

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=i,e^2=h,f^2=g,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, 7999961);