#181048

order	512
nilpotency class	3
isoclinism	isoclinic class of #17411
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1604 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,f^2=i,b^a=be,e^a=eh,f^b=fh,d^c=dhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88244);