#7804848

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

5807 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304331, #72304364, #72304463
maximal normal subgroup of	#72304331, #72304364, #72304463

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7712044);