#465713

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
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

851 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)(7,16)(8,21,23,30)(9,14)(10,15)(11,18)(12,24,29,35)(13,25)(17,28,22,26)(19,27)(20,31)(32,34,36,33);(1,2,3,4)(5,6)(7,11)(8,12)(9,13)(10,15)(14,25)(16,18)(17,26)(19,31)(20,27)(21,35)(22,28)(23,29)(24,30)(32,33)(34,36);(2,4)(5,7,10,20)(6,11,15,27)(8,22,23,17)(9,19,25,18)(12,28,29,26)(13,31,14,16)(21,36,30,32)(24,33,35,34);(5,8,9,24,10,23,25,35)(6,12,13,30,15,29,14,21)(7,17,18,33,20,22,19,34)(11,26,16,32,27,28,31,36)

Polycyclic presentation:

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

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, 372909);