#400947

order	512
nilpotency class	3
isoclinism	isoclinic class of #394483
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2472
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14810
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

824 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:

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

Polycyclic presentation:

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

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, 308143);