#7689888

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4943 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,e^2=g,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, 7597084);