#7687882

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

4547 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7595078);