#213038

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 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)(5,6,10,13)(7,14,12,9)(8,16,18,11)(15,19,17,20)(21,22,25,29,26,24,28,33)(23,27,31,34,30,32,35,36);(5,7,9,6,10,12,14,13)(8,17,19,16,18,15,20,11)(21,23)(22,27)(24,32)(25,31)(26,30)(28,35)(29,34)(33,36);(1,2,3,4)(5,8)(6,11)(7,15)(9,19)(10,18)(12,17)(13,16)(14,20)(23,30)(27,32)(31,35)(34,36);(8,18)(11,16)(15,17)(19,20)(21,24,25,33,26,22,28,29)(23,27,31,34,30,32,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fh,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=chi,d^b=di,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120234);