#7672316

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4606 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)(10,14)(15,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31);(1,3,6,5)(2,4,7,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32);(2,7)(4,8)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31);(1,4,6,8)(2,3,7,5)(17,20,22,19)(18,24,26,23)(21,28,30,27)(25,32,29,31);(1,3,6,5)(2,4,7,8)(9,11,13,12)(10,15,14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30);(1,5,6,3)(2,8,7,4)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,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, 7579512);