#130161

order	512
nilpotency class	4
isoclinism	isoclinic class of #130114
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 546 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37357);