#141002

order	512
nilpotency class	4
isoclinism	isoclinic class of #125534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1533
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=egi,d^2=ghi,e^2=i,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48198);