#222329

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#60694	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

847 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ci,d^b=de,e^b=eh,d^c=di,f^c=fh,g^c=gh,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, 129525);