#141768

order	512
nilpotency class	4
isoclinism	isoclinic class of #12525
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

360 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=ei,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ch

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48964);