#7898230

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

5615 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319623, #72319676, #72319817
maximal normal subgroup of	#72319623, #72319676, #72319817

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7805426);