#7693679

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	#34829	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4735 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,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, 7600875);