#217133

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	#2428	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=ce,d^b=dhi,f^c=fh,g^c=gh

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, 124329);