#6346196

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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	#61447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253392);