#6344749

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
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	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1590 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:

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

Polycyclic presentation:

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

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, 6251945);