#8699104

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

5423 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)(3,4)(5,7)(6,8)(18,22)(19,25)(24,28)(26,27);(1,3,6,5)(2,7,8,4)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,32)(28,29);(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,32)(28,29);(2,8)(3,5)(10,14)(12,15)(17,19,21,25)(18,23,22,30)(20,27,29,26)(24,32,28,31);(1,4)(2,3)(5,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,30)(26,32)(27,31);(1,5,6,3)(2,7,8,4)(9,12,13,15)(10,16,14,11)(17,20,21,29)(18,24,22,28)(19,26,25,27)(23,31,30,32)

Polycyclic presentation:

2^9|b^2=gh,d^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283347, #72312209, #72342324, …
maximal normal subgroup of	#72283347, #72312209, #72342324, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8606300);