#188699

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1066 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)(23,27)(24,30)(25,29)(26,31)(28,32)(33,36)(34,35);(1,2,3,4)(21,23)(22,27)(24,34)(25,32)(26,33)(28,29)(30,35)(31,36);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23)(22,28)(24,32)(25,34)(26,33)(27,29)(30,36)(31,35);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,26,25)(22,29,31,30)(23,32,33,34)(27,35,36,28)

Polycyclic presentation:

2^9|a^2=g,b^2=e,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, 95895);