#7931625

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	#49565	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=ghi,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, 7838821);