#6345184

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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	#60694	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1581 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)(3,11)(4,10)(5,8)(6,9)(7,12)(13,16)(14,15)(17,18,22,25,21,24,30,32)(19,26,28,29,27,23,31,20);(1,3,6,13)(2,7,9,15)(4,12,5,14)(8,11,10,16)(17,19)(18,23)(20,25)(21,27)(22,28)(24,26)(29,32)(30,31);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,11,15,16)(17,20,22,26,21,29,30,23)(18,19,25,28,24,27,32,31);(1,2)(3,7)(4,8)(5,10)(6,9)(11,12)(13,15)(14,16)(19,27)(20,29)(23,26)(28,31);(2,9)(7,15)(8,10)(11,16)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252380);