#149612

order	512
nilpotency class	5
isoclinism	isoclinic class of #13603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#382	C₂³
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2285 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

316 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293179
maximal normal subgroup of	#72293179

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56808);