#400948

order	512
nilpotency class	3
isoclinism	isoclinic class of #394463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,16,11,17,12,27)(7,18,20,29,22,32,21,13)(8,23,25,31,26,14,9,15)(19,30,33,36,34,35,24,28);(5,7,12,20,11,22,10,21)(6,13,27,18,17,29,16,32)(8,24,9,19,26,33,25,34)(14,30,31,36,23,35,15,28);(5,8,10,25,11,26,12,9)(6,14,16,15,17,23,27,31)(7,19,20,33,22,34,21,24)(13,28,18,30,29,36,32,35);(2,4)(5,9,12,26,11,25,10,8)(6,15,27,14,17,31,16,23)(7,19,20,33,22,34,21,24)(13,28,18,30,29,36,32,35)

Polycyclic presentation:

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

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, 308144);