#8375615

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

5648 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342310, #72342324, #72342333
maximal normal subgroup of	#72342310, #72342324, #72342333

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8282811);