#188696

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,8,12,9,11,17,19)(7,13,14,20,15,10,16,18)(21,22)(23,32)(24,28)(25,30)(26,31)(27,35)(29,34)(33,36);(1,2,3,4)(21,23,26,24)(22,28,31,32)(25,34,27,33)(29,30,36,35);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,24,26,23)(22,29,31,36)(25,34,27,33)(28,30,32,35);(21,25,26,27)(22,30,31,35)(23,33,24,34)(28,29,32,36)

Polycyclic presentation:

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

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, 95892);