#7712300

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	#37273	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4918 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619496);