#7855159

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

4802 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)(17,18,23,28)(19,24,31,26)(20,29,30,25)(21,22,27,32);(1,2,5,8)(3,4,7,6)(9,10,13,14)(11,15,12,16)(17,19)(18,24)(20,32)(21,29)(22,30)(23,31)(25,27)(26,28);(1,3)(2,6)(4,8)(5,7)(17,20,23,30)(18,25,28,29)(19,22,31,32)(21,26,27,24);(9,11,13,12)(10,15,14,16)(17,21,23,27)(18,22,28,32)(19,29,31,25)(20,24,30,26);(1,3)(2,4)(5,7)(6,8)(9,12,13,11)(10,16,14,15)(17,19)(18,26)(20,22)(21,29)(23,31)(24,28)(25,27)(30,32);(1,4)(2,7)(3,8)(5,6)(17,22)(18,27)(19,30)(20,31)(21,28)(23,32)(24,25)(26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7762355);