#10250314

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

5728 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342308, #72342316, #72342342
maximal normal subgroup of	#72342308, #72342316, #72342342

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10157510);