#380368

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287564);