#497348

order	512
nilpotency class	4
isoclinism	isoclinic class of #497333
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	#61089	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#576 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1153 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^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,d^b=de,e^b=ei,f^c=fi,g^c=ghi,e^d=ei,g^f=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404544);