#461727

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16400
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

814 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368923);