#7685357

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	#45422	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5158 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)(3,4)(5,7)(6,8)(10,14)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32);(1,3,6,5)(2,4,8,7)(9,10,13,14)(11,15,12,16)(17,19,22,24)(18,23,26,20)(21,27,30,31)(25,29,32,28);(1,3,6,5)(2,7,8,4)(9,11,13,12)(10,15,14,16)(17,18,22,26)(19,20,24,23)(21,25,30,32)(27,28,31,29);(1,4,6,7)(2,3,8,5)(9,11,13,12)(10,15,14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32);(1,3,6,5)(2,4,8,7)(9,11,13,12)(10,15,14,16)(17,20,22,23)(18,24,26,19)(21,29,30,28)(25,27,32,31);(1,5,6,3)(2,7,8,4)(9,12,13,11)(10,16,14,15)(17,21,22,30)(18,25,26,32)(19,27,24,31)(20,28,23,29)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=ghi,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, 7592553);