#10482490

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

5302 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)(3,6)(4,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,31)(28,29)(30,32);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,19,23,22)(18,24,20,26)(21,27,29,30)(25,31,28,32);(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,26,22,24)(21,25,29,28)(27,32,30,31);(1,2)(3,6)(4,7)(5,8)(9,11,12,16)(10,14,13,15)(17,20,23,18)(19,24,22,26)(21,25,29,28)(27,32,30,31);(1,2)(3,5)(4,7)(6,8)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32);(1,3)(2,6)(4,8)(5,7)(17,22,23,19)(18,24,20,26)(21,27,29,30)(25,32,28,31)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265957, #72265960, #72265962
maximal normal subgroup of	#72265957, #72265960, #72265962
maximal quotient of	#72266519, #72266568

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10389686);