#7804854

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

6619 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)(3,8)(4,6)(5,7)(18,20)(22,24);(1,3)(2,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(18,20)(19,23);(3,4)(6,8)(10,14)(12,15)(17,18,21,20)(19,24,23,22);(1,2)(3,8)(4,6)(5,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,24)(21,23);(1,2)(3,8)(4,6)(5,7)(9,11)(10,12)(13,16)(14,15)(17,20,21,18)(19,24,23,22);(1,4,5,3)(2,6,7,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,22)(20,24)(21,23)

Polycyclic presentation:

2^9|c^2=h,e^2=h,f^2=g,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	#72304446, #72304463, #72304464
maximal normal subgroup of	#72304446, #72304463, #72304464

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7712050);