#7920056

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	#37652	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4895 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)(17,18,24,23)(19,28,26,21)(20,25,32,31)(22,27,30,29);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,21)(20,27)(22,25)(23,28)(24,26)(29,32)(30,31);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,30)(22,28)(23,31)(24,32)(26,27);(1,3)(2,6)(4,7)(5,8)(17,21,24,28)(18,26,23,19)(20,30,32,22)(25,27,31,29);(1,4,5,2)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,22,24,30)(18,27,23,29)(19,25,26,31)(20,28,32,21);(1,3)(2,7)(4,6)(5,8)(9,12,13,11)(10,16,14,15)(17,23,24,18)(19,28,26,21)(20,31,32,25)(22,27,30,29)

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=hi,e^2=g,f^2=gh,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, 7827252);