#108255

order	512
nilpotency class	3
isoclinism	isoclinic class of #108238
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1786 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1786 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,7,8,5)(4,6)(9,10,14,19,15,20,13,18)(11,21,23,30,24,34,28,16)(12,17,25,31,26,32,27,33)(22,36,39,29,38,35,37,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16,20,30)(12,22)(13,23)(14,28)(15,24)(17,29,32,40)(18,21,19,34)(25,37)(26,38)(27,39)(31,36,33,35);(9,12)(10,17)(11,22)(13,27)(14,25)(15,26)(16,29)(18,33)(19,31)(20,32)(21,35)(23,39)(24,38)(28,37)(30,40)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,e^b=ei,f^b=fhi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255852, #72255873, #72255890, …
maximal normal subgroup of	#72255852, #72255873, #72255890, …
maximal quotient of	#72266171

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15451);