#6344073

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62800	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2994 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:

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,f^2=i,g^2=hi,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,d^c=dh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251269);