#7854952

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	#36124	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5181 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=h,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, 7762148);