#8108714

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	#39486	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4760 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)(4,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,22,24)(19,25,26,20)(21,29,30,23)(27,28,32,31);(2,4)(6,7)(10,11)(14,15)(19,26)(20,25)(21,30)(23,29);(1,3)(2,6)(4,7)(5,8)(9,10,13,11)(12,14,16,15)(17,19)(18,20)(21,27)(22,26)(23,28)(24,25)(29,31)(30,32);(1,2,5,4)(3,6,8,7)(9,11,13,10)(12,15,16,14)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31);(1,4,5,2)(3,7,8,6)(9,12,13,16)(10,14,11,15)(17,21,22,30)(18,23,24,29)(19,27,26,32)(20,28,25,31);(1,3)(2,6)(4,7)(5,8)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015910);