#7983200

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	#45649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7890396);