#485594

order	512
nilpotency class	4
isoclinism	isoclinic class of #485592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#18152
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

677 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 677 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,g^a=gi,h^a=hi,e^b=eg,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,e^d=ei,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339870
maximal normal subgroup of	#72339870

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392790);