#7806631

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

5379 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327258
maximal normal subgroup of	#72327258
maximal quotient of	#72327465, #72327751, #72328194

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713827);