#8066935

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

5212 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317746, #72317773, #72317873, …
maximal normal subgroup of	#72317746, #72317773, #72317873, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7974131);