#7642941

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5410 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312548, #72342329, #72342331
maximal normal subgroup of	#72312548, #72342329, #72342331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550137);