#405079

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

918 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312275);