#7773563

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	C₂×#540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4640 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,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, 7680759);