#7672319

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	#59137	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579515);