#496016

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21614
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

527 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403212);