#7631182

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	#36434	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5118 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538378);