#8417404

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

6030 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)(3,7)(4,8)(5,6)(11,15)(13,16)(17,18)(19,24)(20,21)(22,23)(25,29)(26,32)(27,30)(28,31);(2,6)(3,8)(9,10,12,14)(11,13,15,16)(18,23)(20,29)(24,31)(26,30);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(21,25)(24,31)(27,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,30)(21,27)(22,28)(23,31)(25,32)(26,29);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,25)(19,26)(21,23)(22,29)(24,27)(28,30)(31,32);(17,21,22,25)(18,20,23,29)(19,27,28,32)(24,30,31,26)

Polycyclic presentation:

2^9|b^2=gh,f^2=gi,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	#72342310, #72342319, #72342342
maximal normal subgroup of	#72342310, #72342319, #72342342

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324600);