#457127

order	512
nilpotency class	3
isoclinism	isoclinic class of #454372
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1146 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 364323);