#6344411

order	512
nilpotency class	3
isoclinism	isoclinic class of #62243
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	C₂²×#41	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1658 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,e^a=ei,f^a=fh,d^c=dhi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251607);