#7708770

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4848 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)(17,18,23,27)(19,26,21,24)(20,22,31,25)(28,30,29,32);(1,2)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,15,16)(17,19,23,21)(18,24,27,26)(20,29,31,28)(22,30,25,32);(2,5)(6,8)(17,20,23,31)(18,25,27,22)(19,28,21,29)(24,30,26,32);(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,29)(21,28)(23,31)(24,30)(25,27)(26,32);(1,2)(3,6)(4,5)(7,8)(17,21,23,19)(18,26,27,24)(20,28,31,29)(22,32,25,30);(1,3)(2,6)(4,7)(5,8)(17,22)(18,20)(19,30)(21,32)(23,25)(24,29)(26,28)(27,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,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, 7615966);