#7734363

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	#36214	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5076 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)(3,5)(4,6)(7,8)(10,13)(14,16)(17,18,23,27)(19,28,21,24)(20,25,29,31)(22,26,32,30);(1,3)(2,7)(4,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19,23,21)(18,24,27,28)(20,26,29,30)(22,31,32,25);(2,4)(3,8)(19,21)(20,29)(24,28)(25,31);(17,20)(18,25)(19,26)(21,30)(22,24)(23,29)(27,31)(28,32);(1,4,6,2)(3,5,8,7)(9,11,12,15)(10,14,13,16)(17,21,23,19)(18,24,27,28)(20,30,29,26)(22,31,32,25);(1,5)(2,3)(4,8)(6,7)(17,22)(18,26)(19,25)(20,24)(21,31)(23,32)(27,30)(28,29)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7641559);