#135490

order	512
nilpotency class	4
isoclinism	isoclinic class of #135469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

551 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=dh,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,f^c=fghi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#135490, #72235853
maximal normal subgroup of	C₂×#135490, #72235853
maximal quotient of	C₂×#135490, #72237326

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42686);