#7737509

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5249 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7644705);