#407080

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1197 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:

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

Polycyclic presentation:

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

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, 314276);