#6459194

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,20,27,21,26,22,23)(19,25,29,31,30,24,28,32);(2,6)(7,8)(18,23)(19,28)(20,22)(24,25)(26,27)(29,30);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,22,28,21,30,20,29)(18,24,23,31,26,25,27,32);(1,4,5,3)(2,8,6,7)(17,19,20,29,21,30,22,28)(18,25,27,31,26,24,23,32);(9,11,12,15)(10,14,13,16)(18,26)(23,27)(24,25)(31,32)

Polycyclic presentation:

2^9|a^2=hi,c^2=fi,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366390);