#457102

order	512
nilpotency class	3
isoclinism	isoclinic class of #457102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1132 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)(3,4)(5,6,11,14,10,16,9,15)(7,17,18,28,21,26,20,12)(8,22,25,30,24,29,23,13)(19,33,32,36,35,27,34,31);(5,7,10,21)(6,12)(8,19,24,35)(9,20,11,18)(13,27)(14,17)(15,26)(16,28)(22,31)(23,34,25,32)(29,36)(30,33);(7,18)(8,23)(9,11)(12,26)(13,22)(14,15)(17,28)(20,21)(24,25)(29,30)(31,36)(32,34);(2,4)(5,8,9,23,10,24,11,25)(6,13,14,22,16,30,15,29)(7,19,20,34,21,35,18,32)(12,27,17,31,28,33,26,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=e,e^2=i,g^2=i,h^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,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=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, 364298);