#7708665

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	#35091	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4960 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:

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

Polycyclic presentation:

2^9|d^2=gh,e^2=h,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, 7615861);