#147025

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

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

726 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)(5,6,11,16,12,18,10,17)(7,19,21,35,22,14,8,23)(9,25,26,34,27,15,28,30)(13,29,32,20,33,24,31,36);(5,7,10,8,12,22,11,21)(6,13,17,32,18,33,16,31)(9,20,26,24,27,36,28,29)(14,34,35,15,19,30,23,25);(1,2,3,4)(5,8,10,22,12,21,11,7)(6,14,16,23,18,19,17,35)(9,24,26,36,27,29,28,20)(13,30,31,15,33,34,32,25)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280195, #72280249, #72280265, …
maximal normal subgroup of	#72280195, #72280249, #72280265, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54221);