#7849928

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	#47794	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5035 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)(17,18,23,24)(19,22,25,31)(20,21,28,30)(26,27,29,32);(1,2)(3,6)(4,7)(5,8)(9,11,13,16)(10,14,12,15)(17,19,23,25)(18,22,24,31)(20,26,28,29)(21,27,30,32);(1,3)(2,5)(4,8)(6,7)(9,10,13,12)(11,14,16,15)(18,24)(19,25)(20,28)(27,32);(9,10,13,12)(11,14,16,15)(17,20)(18,21)(19,26)(22,27)(23,28)(24,30)(25,29)(31,32);(1,3)(2,6)(4,8)(5,7)(9,10,13,12)(11,14,16,15)(17,21)(18,20)(19,27)(22,26)(23,30)(24,28)(25,32)(29,31);(1,2)(3,5)(4,7)(6,8)(9,12,13,10)(11,15,16,14)(17,22)(18,25)(19,24)(20,27)(21,29)(23,31)(26,30)(28,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=g,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, 7757124);