#7854769

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

5779 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,8)(4,6)(5,7)(17,18,22,23)(19,20,25,28)(21,24,30,31)(26,27,32,29);(1,3,5,4)(2,6,7,8)(9,10,13,16)(11,12,15,14)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(18,23)(19,25)(24,31)(26,32);(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32);(1,2)(3,6)(4,8)(5,7)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,29)(22,28)(23,25)(24,32)(26,31)(27,30);(1,4,5,3)(2,6,7,8)(9,12)(10,15)(11,16)(13,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,f^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	#72261985, #72262117, #72262220
maximal normal subgroup of	#72261985, #72262117, #72262220

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7761965);