#188687

order	512
nilpotency class	3
isoclinism	isoclinic class of #161262
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

1062 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)(5,6,9,15,10,14,8,13)(7,11,16,19,17,12,18,20)(21,22)(23,30)(24,28)(25,27)(26,31)(29,33)(32,35)(34,36);(1,2,3,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,14)(13,20)(15,19)(21,23)(22,26)(24,31)(25,32)(27,34)(28,30)(29,35)(33,36);(5,7)(6,12)(8,16)(9,18)(10,17)(11,14)(13,20)(15,19)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(5,8,10,9)(6,13,14,15)(7,16,17,18)(11,19,12,20)(22,28)(23,31)(27,29)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95883);