#402285

order	512
nilpotency class	3
isoclinism	isoclinic class of #396113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

953 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=i,c^2=f,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 309481);