#150215

order	512
nilpotency class	5
isoclinism	isoclinic class of #150210
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

396 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,5)(4,7)(6,8)(9,10,13,17,14,19,28,35)(11,20,23,37,22,30,38,34)(12,16,25,33,26,18,29,36)(15,27,31,24,32,39,40,21);(2,5)(3,8)(6,7)(9,11)(10,15,19,32)(12,24)(13,27)(14,22)(16,20,18,30)(17,34,35,37)(21,26)(23,25)(28,39)(29,38)(31,36,40,33);(9,12,14,26)(10,16,19,18)(11,21,22,24)(13,25,28,29)(15,30,32,20)(17,33,35,36)(23,27,38,39)(31,34,40,37)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=gi,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=ci,d^b=dg,e^b=efg,f^b=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293122, #72293180, #72293203
maximal normal subgroup of	#72293122, #72293180, #72293203

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57411);