#7897674

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	#41951	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4470 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=hi,e^2=h,f^2=gi,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, 7804870);