#6344798

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60676	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1622 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:

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

Polycyclic presentation:

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

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, 6251994);