#111102

order	512
nilpotency class	3
isoclinism	isoclinic class of #9467
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1006 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1006 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240257, #72240335, #72240337
maximal normal subgroup of	#72240257, #72240335, #72240337
maximal quotient of	#72257731, #72257745, #72309245

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18298);