#459442

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1195 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:

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

Polycyclic presentation:

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

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, 366638);