#8810218

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

5548 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249382, #72249522, #72249536
maximal normal subgroup of	#72249382, #72249522, #72249536
maximal quotient of	#72263216

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8717414);