#119422

order	512
nilpotency class	3
isoclinism	isoclinic class of #100117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

832 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=ei

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, 26618);