#7742903

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	#35347	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5614 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)(11,12)(13,15)(14,16)(17,18,23,28)(19,29,30,24)(20,21,26,25)(22,27,32,31);(9,11)(10,14)(12,15)(13,16)(17,19)(18,24)(20,22)(21,31)(23,30)(25,27)(26,32)(28,29);(2,5)(6,7)(10,15)(12,14)(19,30)(20,26)(21,25)(24,29);(1,2)(3,6)(4,5)(7,8)(9,12,13,14)(10,11,15,16)(17,20,23,26)(18,25,28,21)(19,22,30,32)(24,27,29,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,21,23,25)(18,26,28,20)(19,31,30,27)(22,24,32,29);(17,22)(18,27)(19,26)(20,30)(21,24)(23,32)(25,29)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261985, #72262090, #72262230
maximal normal subgroup of	#72261985, #72262090, #72262230

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7650099);