#7654536

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	#35000	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4999 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)(3,8)(4,6)(5,7)(10,13)(14,16)(17,18,22,23)(19,24,26,21)(20,25,29,31)(27,28,32,30);(9,10)(11,14)(12,13)(15,16)(18,23)(19,26)(25,31)(27,32);(1,3,5,4)(2,6,7,8)(17,19)(18,24)(20,27)(21,23)(22,26)(25,28)(29,32)(30,31);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(17,20)(18,25)(19,27)(21,30)(22,29)(23,31)(24,28)(26,32);(1,2)(3,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,24)(18,19,23,26)(20,28,29,30)(25,32,31,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561732);