#404080

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1388 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:

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

Polycyclic presentation:

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

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, 311276);