#226021

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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×C₄⋊C₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

558 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=ci,d^b=de,e^b=eh,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, 133217);