#372906

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

901 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=fg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,e^b=ei,f^c=fi,g^c=gi,e^d=ei

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, 280102);