#8517402

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

5731 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(19,25)(22,29)(26,31)(30,32);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31);(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262211, #72262230
maximal normal subgroup of	#72262211, #72262230

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8424598);