#456906

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1173 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:

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

Polycyclic presentation:

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

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, 364102);