#7727972

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4932 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)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18,23,24)(19,26,20,22)(21,29,30,25)(27,31,28,32);(9,11,12,16)(10,13,14,15)(17,19,23,20)(18,22,24,26)(21,27,30,28)(25,31,29,32);(2,5)(6,7)(18,24)(19,20)(21,30)(31,32);(1,2)(3,6)(4,5)(7,8)(17,20,23,19)(18,22,24,26)(21,28,30,27)(25,31,29,32);(1,3)(2,6)(4,8)(5,7)(17,21,23,30)(18,25,24,29)(19,27,20,28)(22,31,26,32);(17,22)(18,19)(20,24)(21,31)(23,26)(25,27)(28,29)(30,32)

Polycyclic presentation:

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

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, 7635168);