#481482

order	512
nilpotency class	4
isoclinism	isoclinic class of #481477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	#35737	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1224 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1224 conjugacy classes of subgroups — too many to draw. Only its 274 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388678);