#138341

order	512
nilpotency class	4
isoclinism	isoclinic class of #125508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	#32930	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

725 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 725 conjugacy classes of subgroups — too many to draw. Only its 110 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)(5,6)(7,9,11,12)(8,10)(13,14,17,21,15,20,25,27,18,22,16,26,23,28,24,19);(5,7,8,11)(6,9)(10,12)(14,19,22,27)(15,23)(16,24)(17,25)(20,26,28,21);(2,4)(7,11)(9,12)(13,15,18,23)(14,20,22,28)(16,24,17,25)(19,21,27,26)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=ghi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,f^b=fh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45537);