#7672737

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×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5234 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579933);