#512899

order	512
nilpotency class	5
isoclinism	isoclinic class of #512898
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	#61111	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊Q₈ ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1028 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1028 conjugacy classes of subgroups — too many to draw. Only its 112 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306811, #72306817, #72306841
maximal normal subgroup of	#72306811, #72306817, #72306841

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420095);