#7930653

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
prime radical	#38013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4630 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)(9,10)(11,15)(12,14)(13,16)(17,18)(19,21)(20,25)(22,27)(23,28)(24,26)(29,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,26)(18,24,28,21)(20,30,31,27)(22,29,32,25);(10,14)(13,15)(17,20,23,31)(18,25,28,29)(19,27,26,30)(21,22,24,32);(9,10)(11,13)(12,14)(15,16)(17,21,23,24)(18,26,28,19)(20,22,31,32)(25,30,29,27);(9,11)(10,13)(12,16)(14,15)(17,22)(18,27)(19,29)(20,24)(21,31)(23,32)(25,26)(28,30);(1,3,4,7)(2,6,5,8)(17,18)(19,24)(20,25)(21,26)(22,30)(23,28)(27,32)(29,31)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,f^2=ghi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7837849);