#137227

order	512
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 102 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=i,d^2=gi,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44423);