#7927764

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	#41989	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4726 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,e^2=gh,f^2=gh,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, 7834960);