#419789

order	512
nilpotency class	3
isoclinism	isoclinic class of #419788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15273
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338545, #72338547, #72338560
maximal normal subgroup of	#72338545, #72338547, #72338560

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326985);