#8926851

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
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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262230
maximal normal subgroup of	#72262230

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8834047);