#451753

order	512
nilpotency class	3
isoclinism	isoclinic class of #451753
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16012
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339651, #72339677, #72339684
maximal normal subgroup of	#72339651, #72339677, #72339684

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358949);