#10352428

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

5421 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72330757
maximal normal subgroup of	#72330757
maximal quotient of	#72330923, #72331904, #72332054, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10259624);