#7919941

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	#35352	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4893 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)(3,7)(4,5)(6,8)(10,14)(15,16)(17,18,22,26)(19,23,28,31)(20,24,27,30)(21,25,29,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,30)(26,31);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,20,22,27)(18,24,26,30)(19,21,28,29)(23,25,31,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,16,14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(1,4,5,2)(3,7,8,6)(17,19)(18,23)(20,21)(22,28)(24,25)(26,31)(27,29)(30,32);(1,3)(2,7)(4,6)(5,8)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7827137);