#7687937

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5075 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=i,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, 7595133);