#7812177

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	#35347	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5714 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7719373);