#222308

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	#3012	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

887 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)(17,18,21,26,22,19,27,25)(20,24,31,28,32,29,30,23);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19)(18,22)(20,23)(21,25)(24,31)(26,27)(28,32)(29,30);(3,9)(6,12)(10,15)(13,16)(17,20)(18,23)(19,28)(21,30)(22,32)(24,25)(26,29)(27,31);(2,7)(6,12)(8,14)(13,16)(17,20)(18,24)(19,29)(21,31)(22,32)(23,25)(26,28)(27,30)

Polycyclic presentation:

2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,c^b=ci,d^b=de,e^b=eh,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, 129504);