#7857087

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	#44886	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4995 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=gh,d^2=h,e^2=h,f^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, 7764283);