#8934620

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8026 conjugacy classes of subgroups; only the 2910 normal subgroups are shown.

normal subgroups only

This group has 8026 conjugacy classes of subgroups — too many to draw. Only its 2910 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) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(2,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,20,22)(19,23,24,21);(3,7)(6,8)(11,16)(13,14)(17,18)(19,21)(20,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,22)(21,24);(9,11,12,16)(10,14,15,13)(17,18,20,22)(19,21,24,23);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,24)(22,23)

Polycyclic presentation:

2^9|a^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227582, #72227584, #72227585, …
Fitting subgroup of	#72227582, #72227936, #72228061, …
maximal subgroup of	#72213214, #72213228, #72213244, …
maximal normal subgroup of	#72213214, #72213228, #72213244, …
maximal quotient of	#72235019, #72268012, #72268079, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841816);