#138088

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	#2619	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 408 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,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, 45284);