#7696349

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4341 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,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, 7603545);