#8115334

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
prime radical	#37670	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4563 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022530);