#118679

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

928 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25875);