#511805

order	512
nilpotency class	4
isoclinism	isoclinic class of #511802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#23909	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2324 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 2324 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,17)(8,14)(9,23)(10,13)(11,16)(12,19)(15,26)(18,29)(20,33)(21,32)(22,30)(24,31)(25,34)(27,36)(28,35);(5,7,11,21)(6,12,16,29)(8,19,9,18)(10,25,26,20)(13,27,15,28)(14,32,23,17)(22,35,24,36)(30,34,31,33);(6,13)(10,26)(12,27)(14,30)(15,16)(17,33)(20,25)(22,24)(23,31)(28,29)(32,34)(35,36);(2,4)(5,8,11,9)(6,14,16,23)(7,18,21,19)(10,22,26,24)(12,17,29,32)(13,30,15,31)(20,35,25,36)(27,33,28,34)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279992, #72280283
maximal normal subgroup of	#72279992, #72280283

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419001);