#7717225

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

4956 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(9,11,12,16)(10,13,14,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(2,6)(3,7)(18,22)(19,26)(24,28)(27,32);(1,3,5,7)(2,4,6,8)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,4)(2,7)(3,6)(5,8)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7624421);