#7919989

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	#36428	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5355 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7827185);