#7695722

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4533 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7602918);