#451308

order	512
nilpotency class	3
isoclinism	isoclinic class of #428398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2609
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15994
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

752 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358504);