#94358

order	512
nilpotency class	5
isoclinism	isoclinic class of #94343
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
prime radical	C₂³	#358
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

150 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 256

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=i,e^2=fgi,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225461, #72225521, #72225531
maximal normal subgroup of	#72225461, #72225521, #72225531
maximal quotient of	#72225456

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1554);