#8075300

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

5159 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)(3,6)(4,5)(7,8)(10,14)(15,16)(17,18)(19,26)(20,22)(21,25)(23,27)(24,30)(28,29)(31,32);(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,31)(21,28)(22,29)(23,30)(25,32)(26,27);(2,5)(6,8)(17,20,23,21)(18,22,27,25)(19,28,30,31)(24,32,26,29);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,21,23,20)(18,25,27,22)(19,28,30,31)(24,32,26,29);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,16,14,15)(17,22,23,25)(18,21,27,20)(19,29,30,32)(24,28,26,31);(1,2)(3,6)(4,5)(7,8)(17,19)(18,26)(20,28)(21,31)(22,29)(23,30)(24,27)(25,32)

Polycyclic presentation:

2^9|c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311241, #72311295, #72311493, …
maximal normal subgroup of	#72311241, #72311295, #72311493, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982496);