#380137

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1163 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,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, 287333);