#190621

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1006 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)(11,14,15,16)(17,18,21,24,20,23,28,31)(19,25,27,32,26,22,29,30);(2,5)(6,8)(9,10,13,12)(11,14,15,16)(17,19,20,26)(18,22,23,25)(21,27,28,29)(24,30,31,32);(9,11)(10,14)(12,16)(13,15)(18,23)(21,28)(22,25)(27,29);(1,3,4,7)(2,6,5,8)(9,10,13,12)(11,14,15,16)(18,23)(22,25)(24,31)(30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=ehi,f^2=i,g^2=h,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, 97817);