#7855627

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	#39357	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4834 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)(17,18,23,25)(19,24,28,31)(20,22,27,32)(21,30,29,26);(1,2,5,8)(3,4,7,6)(9,10,13,14)(11,15,12,16)(17,19)(18,24)(20,29)(21,27)(22,26)(23,28)(25,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(18,25)(19,28)(20,27)(26,30);(17,20,23,27)(18,22,25,32)(19,29,28,21)(24,26,31,30);(1,3)(2,4)(5,7)(6,8)(9,11,13,12)(10,15,14,16)(17,21,23,29)(18,26,25,30)(19,27,28,20)(22,31,32,24);(1,4)(2,7)(3,8)(5,6)(9,12,13,11)(10,16,14,15)(17,22)(18,27)(19,26)(20,25)(21,24)(23,32)(28,30)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^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, 7762823);