#111471

order	512
nilpotency class	3
isoclinism	isoclinic class of #9974
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#3186	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1292 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1292 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,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, 18667);