#6468674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	#35276	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1623 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375870);