#188702

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

938 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)(5,6,9,12,10,13,8,17)(7,14,15,19,16,11,18,20)(21,22)(23,33)(24,28)(25,32)(26,31)(27,30)(29,35)(34,36);(1,2,3,4)(21,23,26,24)(22,28,31,33)(25,35,27,34)(29,32,36,30);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,24,26,23)(22,29,31,36)(25,35,27,34)(28,32,33,30);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,26,27)(22,30,31,32)(23,34,24,35)(28,36,33,29)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,d^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,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, 95898);