#7967796

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
prime radical	#36152	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241096, #72241103, #72241105
maximal normal subgroup of	#72241096, #72241103, #72241105

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7874992);