#7876726

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

5701 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)(9,10)(11,14)(12,15)(13,16)(17,18,21,24)(19,22,20,23);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24);(1,2)(3,8)(4,6)(5,7)(9,10,13,16)(11,12,15,14);(9,11)(10,14)(12,16)(13,15)(17,19,21,20)(18,22,24,23);(1,4,5,3)(2,6,7,8)(9,11)(10,12)(13,15)(14,16)(17,20,21,19)(18,23,24,22);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,16)(13,14)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304331, #72304392, #72304435
maximal normal subgroup of	#72304331, #72304392, #72304435

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7783922);