#8075498

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	#43212	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4637 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=h,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7982694);