#511859

order	512
nilpotency class	4
isoclinism	isoclinic class of #511859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32899
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #519 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2146 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 2146 conjugacy classes of subgroups — too many to draw. Only its 184 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)(4,7)(5,8)(6,12)(9,16)(10,17)(11,19)(14,25)(15,26)(18,21)(20,23)(22,27)(24,29)(28,32)(30,31);(2,7)(4,17)(5,19)(8,21)(9,22)(10,12)(11,23)(13,25)(14,27)(15,29)(24,31)(26,32);(2,8)(5,19)(7,21)(9,24)(10,23)(11,12)(13,26)(15,29)(18,20)(22,31)(25,32)(28,30);(1,3)(2,9,12,13)(4,14)(5,15)(6,16)(7,22,10,25)(8,24,11,26)(17,27)(18,28)(19,29)(20,30)(21,31,23,32)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260900, #72261089, #72261130
maximal normal subgroup of	#72260900, #72261089, #72261130

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419055);