#222687

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#61449	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

881 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=chi,d^b=de,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 129883);