#456894

order	512
nilpotency class	3
isoclinism	isoclinic class of #453991
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

1221 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)(5,6,11,9)(7,15,19,30)(8,20,23,35)(10,25,27,14)(12,26,17,18)(13,29,24,22)(16,32,21,28)(31,36,34,33);(1,3,4,2)(5,7,11,19)(6,12,9,17)(8,21,23,16)(10,26,27,18)(13,28,24,32)(14,30,25,15)(20,34,35,31)(22,33,29,36);(2,3)(5,8,10,22,11,23,27,29)(6,13,14,20,9,24,25,35)(7,16,18,33,19,21,26,36)(12,28,15,31,17,32,30,34);(5,9,11,6)(7,17,19,12)(8,20,23,35)(10,14,27,25)(13,29,24,22)(15,26,30,18)(16,31,21,34)(28,36,32,33)

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,d^2=i,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,e^b=ei,f^b=fi,h^b=hi,d^c=de,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, 364090);