#7678587

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	#59137	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4598 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)(3,4)(5,8)(6,7)(10,14)(15,16)(19,22)(24,26)(27,28)(31,32);(1,3,6,5)(2,4,7,8)(9,10,13,14)(11,15,12,16)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,31)(29,30);(2,7)(4,8)(17,19,23,22)(18,24,21,26)(20,28,30,27)(25,32,29,31);(1,4,6,8)(2,3,7,5)(17,20,23,30)(18,25,21,29)(19,27,22,28)(24,31,26,32);(9,11,13,12)(10,15,14,16)(17,21,23,18)(19,24,22,26)(20,29,30,25)(27,31,28,32);(1,5,6,3)(2,8,7,4)(9,12,13,11)(10,16,14,15)(17,22,23,19)(18,24,21,26)(20,28,30,27)(25,31,29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585783);