#402151

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

781 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)(5,6,10,18,11,17,12,16)(7,15,9,13,20,28,21,27)(8,14,23,32,25,33,24,31)(19,30,22,26,35,29,34,36);(5,7,12,21,11,20,10,9)(6,13,18,28,17,27,16,15)(8,22,23,35,25,34,24,19)(14,29,31,26,33,30,32,36);(5,8,10,23,11,25,12,24)(6,14,16,31,17,33,18,32)(7,19,9,22,20,35,21,34)(13,26,15,30,27,36,28,29);(2,4)(5,9,10,20,11,21,12,7)(6,15,16,27,17,28,18,13)(8,22,23,35,25,34,24,19)(14,30,31,36,33,29,32,26)

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=e,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^c=fi,g^c=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, 309347);