#7857733

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	#38020	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5132 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7764929);