#105604

order	512
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁶	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1802 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1802 conjugacy classes of subgroups — too many to draw. Only its 314 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,f^b=fh,f^c=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12800);