#7753416

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	#43019	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5022 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7660612);