#7708376

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	#37436	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7615572);