#7645357

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	#36706	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5032 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7552553);