#8496821

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

4872 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)(5,6)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,23,22)(19,24,29,28)(20,30,21,25)(26,32,27,31);(2,6)(3,8)(10,13)(14,15)(17,19,23,29)(18,24,22,28)(20,27,21,26)(25,32,30,31);(9,10)(11,14)(12,13)(15,16)(17,20,23,21)(18,25,22,30)(19,26,29,27)(24,31,28,32);(2,6)(4,7)(9,11)(10,14)(12,16)(13,15)(17,21,23,20)(18,25,22,30)(19,27,29,26)(24,31,28,32);(1,3)(2,7)(4,6)(5,8)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,31)(30,32);(1,4,5,7)(2,3,6,8)(17,22,23,18)(19,28,29,24)(20,30,21,25)(26,32,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308965, #72308996
maximal normal subgroup of	#72308965, #72308996

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8404017);