#8093094

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3926 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^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, 8000290);