#121349

order	512
nilpotency class	3
isoclinism	isoclinic class of #10026
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 596 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,e^b=ei,f^b=fh,d^c=di,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, 28545);