#8898087

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

6716 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227654, #72227667, #72227753, …
Fitting subgroup of	#72227654, #72227662, #72227671, …
maximal subgroup of	#72227654, #72227660, #72227665, …
maximal normal subgroup of	#72227654, #72227660, #72227665, …
maximal quotient of	#72267914, #72268012, #72268220, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8805283);