#8406567

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

5403 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,20,25,28)(21,29,32,24)(26,31,30,27);(2,3)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,20)(21,30)(22,25)(23,28)(24,31)(26,32)(27,29);(1,2,5,3)(4,6,8,7)(9,10)(11,13)(12,14)(15,16)(18,23)(19,25)(24,29)(26,30);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,20)(18,19)(21,31)(22,28)(23,25)(24,30)(26,29)(27,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,27)(22,32)(23,29)(25,30)(28,31);(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265961, #72265967
maximal normal subgroup of	#72265961, #72265967
maximal quotient of	#72266563

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313763);