#147486

order	512
nilpotency class	4
isoclinism	isoclinic class of #13030
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dhi,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250430, #72250433, #72250445, …
maximal normal subgroup of	#72250430, #72250433, #72250445, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54682);