#511816

order	512
nilpotency class	4
isoclinism	isoclinic class of #511816
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	C₂²×Q₈	C₂⁴
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2374 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 2374 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)(4,10)(5,18)(6,9)(7,12)(8,14)(11,21)(15,28)(16,23)(17,25)(19,24)(20,26)(22,32)(27,30)(29,31);(1,3)(2,8)(4,14)(5,19)(6,15)(7,16)(9,22)(10,23)(11,27)(12,24)(13,18)(17,30)(20,32)(21,31)(25,28)(26,29);(2,9)(4,5)(6,21)(8,22)(10,26)(11,12)(13,28)(14,19)(15,31)(18,25)(23,29)(24,27);(1,4)(2,10)(3,14)(5,7)(6,17)(8,23)(9,25)(11,26)(12,18)(13,24)(15,30)(16,19)(20,21)(22,28)(27,29)(31,32)

Polycyclic presentation:

2^9|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,g^b=gi,d^c=di,e^c=eh,f^c=fi,e^d=ei,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261130, #72261203, #72283424
maximal normal subgroup of	#72261130, #72261203, #72283424

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419012);