#7651238

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

5166 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)(6,8)(10,13)(11,15);(1,2)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18,22,23)(19,24,28,31)(20,21,25,30)(26,32,29,27);(9,10,12,13)(11,14,15,16)(18,23)(21,30)(24,31)(27,32);(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,26)(30,32);(1,3,4,7)(2,6,5,8)(17,20,22,25)(18,21,23,30)(19,26,28,29)(24,32,31,27);(17,21)(18,25)(19,27)(20,23)(22,30)(24,26)(28,32)(29,31)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,e^2=ghi,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	#72319686, #72319769
maximal normal subgroup of	#72319686, #72319769

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7558434);