#453815

order	512
nilpotency class	3
isoclinism	isoclinic class of #453815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16125
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

734 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361011);