#7645386

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	#37439	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5002 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)(3,7,8,6)(10,14)(15,16)(17,18)(19,28)(20,26)(21,23)(22,25)(24,27)(29,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,10,13,14)(11,15,12,16)(17,19,24,22)(18,25,27,28)(20,31,21,29)(23,30,26,32);(1,3)(2,6)(4,7)(5,8)(17,20,24,21)(18,26,27,23)(19,29,22,31)(25,30,28,32);(2,4)(6,7)(9,11,13,12)(10,15,14,16)(17,21,24,20)(18,23,27,26)(19,29,22,31)(25,30,28,32);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,16,14,15)(17,22,24,19)(18,25,27,28)(20,31,21,29)(23,32,26,30);(1,3)(2,6)(4,7)(5,8)(17,23)(18,20)(19,30)(21,27)(22,32)(24,26)(25,31)(28,29)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7552582);