#7855818

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

5407 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,e^2=gh,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	#72312307, #72342294, #72342313
maximal normal subgroup of	#72312307, #72342294, #72342313

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7763014);