#497700

order	512
nilpotency class	3
isoclinism	isoclinic class of #359901
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2342
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#22554
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

648 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=deh,e^b=ei,g^b=gi,d^c=di,g^c=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404896);