#137143

order	512
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×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	C₂×#1999	C₂
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

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

normal subgroups only

This group has 946 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) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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, 44339);