#433302

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15476
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1067 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285906, #72286050, #72286206, …
maximal normal subgroup of	#72285906, #72286050, #72286206, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340498);