#511837

order	512
nilpotency class	4
isoclinism	isoclinic class of #511835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32898
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2307 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 2307 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,13)(4,17)(5,20)(6,23)(7,12)(8,14)(9,16)(10,18)(11,15)(19,26)(21,30)(22,28)(24,27)(25,31)(29,32);(1,3,8,7)(2,9,14,17)(4,18,16,6)(5,15,22,24)(10,12,23,13)(11,25,27,30)(19,31,29,21)(20,32,28,26);(1,4,8,16)(2,10,14,23)(3,6,7,18)(5,19,22,29)(9,13,17,12)(11,26,27,32)(15,21,24,31)(20,25,28,30);(1,5)(2,11)(3,15)(4,19)(6,21)(7,24)(8,22)(9,25)(10,26)(12,20)(13,28)(14,27)(16,29)(17,30)(18,31)(23,32)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283528
maximal normal subgroup of	#72283528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419033);