#190697

order	512
nilpotency class	3
isoclinism	isoclinic class of #160780
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

1138 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)(9,10,13,15,11,14,12,16)(17,18,21,24,20,22,28,30)(19,25,27,32,26,23,29,31);(2,5)(6,8)(10,14)(15,16)(18,22)(23,25)(24,30)(31,32);(10,15)(12,13)(14,16)(17,19,20,26)(18,23,22,25)(21,29,28,27)(24,32,30,31);(1,3,4,7)(2,6,5,8)(18,22)(23,25)(24,30)(31,32)

Polycyclic presentation:

2^9|a^2=g,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, 97893);