#138089

order	512
nilpotency class	4
isoclinism	isoclinic class of #125092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45285);