#7708522

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4722 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)(9,10,12,14)(11,15,16,13)(19,26)(21,29)(23,24)(25,28);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,18,22,20)(19,24,26,23)(21,25,29,28)(27,31,32,30);(2,5)(6,8)(17,19,22,26)(18,23,20,24)(21,27,29,32)(25,30,28,31);(17,20,22,18)(19,23,26,24)(21,28,29,25)(27,30,32,31);(1,2)(3,6)(4,5)(7,8)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^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, 7615718);