#461160

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

870 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)(5,6,12,27,10,16,11,17)(7,13,14,28,19,8,20,22)(9,23,26,32,24,15,25,33)(18,21,31,35,34,29,30,36);(5,7,10,19)(6,13)(8,16)(9,18,24,34)(11,14,12,20)(15,29)(17,28)(21,23)(22,27)(25,31,26,30)(32,36)(33,35);(5,8)(6,14,16,20)(7,17,19,27)(9,21)(10,13)(11,22)(12,28)(15,31,23,30)(18,32,34,33)(24,29)(25,35)(26,36);(2,4)(5,9,10,24)(6,15,16,23)(7,18,19,34)(8,21,13,29)(11,26,12,25)(14,30,20,31)(17,33,27,32)(22,36,28,35)

Polycyclic presentation:

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

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, 368356);