#497352

order	512
nilpotency class	4
isoclinism	isoclinic class of #497339
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#61082	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#576 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1125 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=ei,f^c=fi,g^c=ghi,e^d=ei,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404548);