#7966958

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	#39135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4814 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7874154);