#7651233

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

5860 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)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,25,27,21)(20,24,30,31)(26,32,28,29);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,21)(20,26)(22,27)(23,25)(24,29)(28,30)(31,32);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,16)(14,15)(18,23)(21,25)(24,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,21)(20,28)(22,27)(23,25)(24,32)(26,30)(29,31);(17,20,22,30)(18,24,23,31)(19,26,27,28)(21,29,25,32);(17,21)(18,19)(20,29)(22,25)(23,27)(24,26)(28,31)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312548, #72342319, #72342320
maximal normal subgroup of	#72312548, #72342319, #72342320

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7558429);