#7696412

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	#52307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4661 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7603608);