#135208

order	512
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

660 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=d,e^2=i,f^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gi,e^c=ei,f^c=fgi,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267644
maximal normal subgroup of	#72267644
maximal quotient of	#72267647

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42404);