#183642

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

742 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=h,f^2=i,b^a=be,d^a=dh,d^b=di,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90838);