#130577

order	512
nilpotency class	4
isoclinism	isoclinic class of #130512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	#2535	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2356 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

538 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ghi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^c=fgi,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37773);