#465715

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

850 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 372911);