#222676

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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	#61431	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

894 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:

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

Polycyclic presentation:

2^9|a^2=g,b^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=ci,d^b=de,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 129872);