#147022

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1799
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

756 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)(7,18)(8,22)(9,24,26,15)(10,28)(11,16)(12,17)(13,19,31,34)(14,21)(20,33)(23,30,35,36)(25,32,27,29);(5,7,11,20,12,21,10,8)(6,13,16,30,17,31,28,36)(9,19,27,35,26,34,25,23)(14,29,22,15,18,32,33,24);(1,2,3,4)(5,8,10,21,12,20,11,7)(6,14,16,22,17,18,28,33)(9,23,25,34,26,35,27,19)(13,29,30,15,31,32,36,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240717, #72240800, #72240802, …
maximal normal subgroup of	#72240717, #72240800, #72240802, …
maximal quotient of	#72240721

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54218);