#8066887

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

5235 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318787, #72319870, #72319886
maximal normal subgroup of	#72318787, #72319870, #72319886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7974083);