#7684392

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

5638 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304331, #72304334, #72304455
maximal normal subgroup of	#72304331, #72304334, #72304455

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7591588);