#511818

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 ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2315 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 2315 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)(3,14)(4,12)(5,11)(6,20)(7,22)(8,13)(9,15)(10,19)(16,31)(17,28)(18,27)(21,26)(23,24)(25,32)(29,30);(1,3)(2,9)(4,15)(5,16)(6,21)(7,23)(8,17)(10,24)(11,25)(12,28)(13,26)(14,20)(18,30)(19,32)(22,29)(27,31);(1,4,8,6)(2,10,13,27)(3,15,17,21)(5,19,7,18)(9,24,26,31)(11,20,22,12)(14,29,28,25)(16,32,23,30);(1,5,8,7)(2,11,13,22)(3,16,17,23)(4,18,6,19)(9,25,26,29)(10,12,27,20)(14,24,28,31)(15,30,21,32)

Polycyclic presentation:

2^9|c^2=i,d^2=i,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	#72261107, #72261203, #72283075
maximal normal subgroup of	#72261107, #72261203, #72283075

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419014);