#7633308

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	#35293	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5238 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540504);