#8698908

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

5147 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,e^2=g,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	#72311171, #72311201, #72311959, …
maximal normal subgroup of	#72311171, #72311201, #72311959, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8606104);