#7744975

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4769 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:

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

Polycyclic presentation:

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

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, 7652171);