#186696

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2450 conjugacy classes of subgroups; only the 370 normal subgroups are shown.

normal subgroups only

This group has 2450 conjugacy classes of subgroups — too many to draw. Only its 370 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 — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93892);