#7684725

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4680 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)(3,7)(4,8)(5,6)(10,14)(15,16)(17,18)(19,28)(20,25)(21,26)(22,27)(23,24)(29,31)(30,32);(9,10)(11,15)(12,16)(13,14)(17,19,23,27)(18,22,24,28)(20,21,29,32)(25,30,31,26);(2,6)(7,8)(9,11,13,12)(10,15,14,16)(18,24)(19,27)(20,29)(26,30);(1,2)(3,7)(4,8)(5,6)(9,11,13,12)(10,15,14,16)(17,20,23,29)(18,25,24,31)(19,21,27,32)(22,30,28,26);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,15,14,16)(17,21)(18,26)(19,29)(20,27)(22,25)(23,32)(24,30)(28,31);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,16,14,15)(17,22)(18,27)(19,24)(20,30)(21,31)(23,28)(25,32)(26,29)

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=g,e^2=gh,f^2=gh,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, 7591921);