#190729

order	512
nilpotency class	3
isoclinism	isoclinic class of #160780
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

960 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=ehi,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fh,f^c=fi,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97925);