#222663

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	#61434	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

862 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:

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

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,e^a=eh,f^a=fi,c^b=ci,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, 129859);